collateral equilibrium, i: a basic framework by john

COLLATERAL EQUILIBRIUM, I: A BASIC FRAMEWORK

By

John Geanakoplos and William R. Zame

COWLES FOUNDATION PAPER NO. 1431

COWLES FOUNDATION FOR RESEARCH IN ECONOMICS YALE UNIVERSITY

Box 208281 New Haven, Connecticut 06520-8281

2014

http://cowles.econ.yale.edu/

http://cowles.econ.yale.edu/

Econ Theory (2014) 56:443–492DOI 10.1007/s00199-013-0797-4

RESEARCH ARTICLE

Collateral equilibrium, I: a basic framework

John Geanakoplos · William R. Zame

Received: 13 August 2013 / Accepted: 10 December 2013 / Published online: 31 December 2013© Springer-Verlag Berlin Heidelberg 2013

Abstract Much of the lending in modern economies is secured by some form ofcollateral: residential and commercial mortgages and corporate bonds are familiarexamples. This paper builds an extension of general equilibrium theory that incorpo-rates durable goods, collateralized securities, and the possibility of default to arguethat the reliance on collateral to secure loans and the particular collateral requirementschosen by the social planner or by the market have a profound impact on prices, alloca-tions, market structure, and the efficiency of market outcomes. These findings provideinsights into housing and mortgage markets, including the subprime mortgage market.

Earlier versions of this paper circulated under the titles “Collateral, Default and Market Crashes” and“Collateral and the Enforcement of Intertemporal Contracts.” We thank Pradeep Dubey, several refereesand editors, and seminar audiences at British Columbia, Caltech, Harvard, Illinois, Iowa, Minnesota, PennState, Pittsburgh, Rochester, Stanford, UC Berkeley, UCLA, USC, Washington University in St. Louis,the NBER Conference Seminar on General Equilibrium Theory the Stanford Institute for TheoreticalEconomics for comments, and Daniele Terlizzese for correcting errors in Example 2. Financial supportwas provided by the Cowles Foundation (Geanakoplos), the John Simon Guggenheim MemorialFoundation (Zame), the UCLA Academic Senate Committee on Research (Zame), the National ScienceFoundation (Geanakoplos, Zame), and the Einaudi Institute for Economics and Finance (Zame). Anyopinions, findings, and conclusions or recommendations expressed in this material are those of the authorsand do not necessarily reflect the views of any funding agency.

J. GeanakoplosSanta Fe Institute, Santa Fe, NM, USA

J. GeanakoplosYale University, New Haven, CT, USA

W. R. Zame (B)UCLA, Los Angeles, CA, USAe-mail: [email protected]

W. R. ZameEIEF, Rome, Italy

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444 J. Geanakoplos, W. R. Zame

Keywords Collateral · Default · GEI

JEL Classification D5

1 Introduction

Recent events in financial markets provide a sharp reminder that much of the lending inmodern economies is secured by some form of collateral: residential and commercialmortgages are secured by the mortgaged property itself, corporate bonds are securedby the physical assets of the firm, and collateralized mortgage obligations and debtobligations and other similar instruments are secured by pools of loans that are in turnsecured by physical property. The total of such collateralized lending is enormous:in 2007, the value of US residential mortgages alone was roughly $10 trillion, andthe (notional) value of collateralized credit default swaps was estimated to exceed$50 trillion. The reliance on collateral to secure loans is so familiar that it might beeasy to forget that it is a relatively recent innovation: extra-economic penalties suchas debtor’s prisons, indentured servitude, and even execution were in widespread usein Western societies into the middle of the nineteenth century.

Reliance on collateral to secure loans—rather than on extra-economic penalties—avoids the moral and ethical issues of imposing penalties in the event of bad luck,the cost of imposing penalties, and the difficulty in finding the defaulter in order toimpose penalties at all. Penalties represent a pure deadweight loss: to the borrowerwho suffers the penalty and to the society as a whole in administering it. The relianceon collateral, which simply transfers resources from one owner to another, is intendedto avoid (some of) this deadweight loss.1 However, as this paper argues, the relianceon collateral to secure loans can have a profound effect on prices, on allocations, on thestructure of financial institutions, and especially on the efficiency of market outcomes.

To make these points, we formulate an extension of intertemporal general equilib-rium theory that incorporates durable goods (physical or financial assets), collateral,and the possibility of default. To focus the discussion, we restrict attention to a pureexchange framework with two dates but many possible states of nature (representingthe uncertainty at time 0 about exogenous shocks at time 1). As usual in general equi-librium theory, we view individuals as anonymous price-takers; for simplicity, we usea framework with a finite number of agents and divisible loans.2 Central to the modelis that the definition of a security must now include not just its promised deliveriesbut also the collateral required to back that promise; the same promise backed by adifferent collateral constitutes a different security and might trade at a different price,because it might give rise to different realized deliveries. We assume that collateral

1 In practice, seizure of collateral may involve deadweight losses of its own.2 Anonymity and price-taking might appear strange in an environment in which individuals might default.In our context, however, individuals will default when the value of promises exceeds the value of collateraland not otherwise; thus, lenders do not care about the identity of borrowers, but only about the collateral theybring. The assumption of price-taking might be made more convincing by building a model that incorporatesa continuum of individuals, and the realism of the model might be enhanced by allowing for indivisibleloans, but doing so would complicate the model without qualitatively changing the conclusions.

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Collateral equilibrium, I 445

is held and used by the borrower and that forfeiture of collateral is the only conse-quence of default; in particular, there are no penalties for default other than forfeitureof the collateral, and there is no destruction of property in the seizure of collateral.As a result, borrowers will always deliver the minimum of what is promised and thevalue of the collateral. Lenders, knowing this, need not worry about the identity of theborrowers but only about the future value of the collateral. Our model requires thateach security be collateralized by a distinct bundle of assets (usually physical goods);residential mortgages (in the absence of second liens) provide the canonical exampleof such securities.3 Although default is suggestive of disequilibrium, our model passesthe basic test of consistency: under the hypotheses on agent behavior and foresightthat are standard in the general equilibrium literature, equilibrium always exists (The-orem 1). The existence of equilibrium rests on the fact that collateral requirementsplace an endogenous bound on both long and short sales. (The reader will recall that itis the possibility of unbounded short sales that leads to non-existence of equilibriumin the standard model of general equilibrium with incomplete markets (GEI). See thediscussion following Theorem 1.)

The familiar models of Walrasian equilibrium (WE) and of GEI tacitly assume thatall agents keep all their promises, but ignore the question of why agents should keeptheir promises; implicitly these models assume that there are infinite penalties forbreaking promises—so that agents always keep the promises they make and alwaysmake only promises that they will be able to keep. Our model of collateral equilibrium(CE) makes explicit the reasons why agents do or do not keep their promises and door do not make promises that they will not be able to keep—and the reasons why otheragents accept these promises, even knowing they may not be kept.

We show (modulo some technical differentiability and interiority assumptions) thatwhenever CE diverges from GEI, some agent would have borrowed more at the pre-vailing interest rates if he did not have to put up the collateral to get the loan but still(miraculously) had to maintain the same delivery rates.4 Credit constraints are thedistinguishing characteristic of CE. Somewhat more surprisingly, we show that thereis a second distinguishing characteristic of CE: some durable good must trade for aprice that is strictly higher than its marginal utility to some agent.5 When collateralmatters, it creates both price and consumption distortions of a particular kind. Weidentify the deviation in commodity prices as a “collateral value,” which leads to com-modity prices that are always at least as high as fundamental values and sometimesstrictly higher, and the deviation in security prices as a “liquidity value,” which leadsto security prices that are always at least as high as fundamental values and sometimesstrictly higher (Theorem 2).

3 Geanakoplos and Zame (2013) expands the model to include a broader range of collateralized assets,including pools.4 In other words, he would have sold more securities at the going prices if he were freed from the burdenof posting collateral.5 The agent who did not borrow as much as he would have at the going prices if he did not have to put upcollateral (as in GEI) does not do so because the collateral he needs to post trades for a price that exceedsits marginal utility to him.

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Collateral and liquidity values have important implications for pricing and produc-tion: securities with the same deliveries can sell for different prices (so buyers maynot earn the standard “market risk-adjusted” rate of return), and production may bedistorted (compared to first best) toward goods that can be used as collateral and awayfrom goods that cannot. They also have important implications for the structure offinancial markets: various promises must compete for the underlying collateral, butonly those securities that create the maximal liquidity value, equal to the collateralvalue of the underlying collateral, will be sold; other promises, which might havebrought greater welfare gains if they were traded (and miraculously delivered with-out benefit of collateral), will not be traded because they waste collateral. In extremecases, financial markets may shut down entirely if agents who want/need to borrow andwould be happy to do so at prevailing interest rates are discouraged from borrowingbecause they do not value the collateral enough that they are willing to hold it. In aslightly different vein, whenever CE diverges from WE there must also be divergencefrom Pareto optimality: CE that are Pareto optimal are necessarily WE (Theorem 3).

These ideas are illustrated in several simple examples. In Example 1 (a mortgagemarket with no uncertainty), we compute CE as a function of the wealth distributionand collateral requirement and identify parameter regions where CE is Pareto opti-mal and coincident with WE/GEI and parameter regions where it is not; in the latterregions, we identify the distortions that are present. We find that the asset price ofthe collateral is much more sensitive to the distribution of wealth at time 0 in CEthan in WE. The price is also very sensitive to exogenously imposed collateral (lever-age) requirements. The welfare impact of collateral requirements is ambiguous: lowercollateral requirements make it possible for buyers to hold more houses but createmore competition for the same houses, thereby driving up the prices.6,7 In Exam-ples 2 and 3, we add uncertainty to the basic mortgage market to examine the effectsof potential and actual default on outcomes, on welfare, and on the market structure.Surprisingly, we find that collateral requirements that lead to default in equilibriummay (ex ante) Pareto dominate collateral requirements that do not lead to default;moreover, such collateral requirements may be endogenously chosen by the market.This suggests an important implication for the subprime mortgage market: even if it istrue that defaults on subprime mortgages led to a crash ex post, such mortgages mighthave been Pareto improving ex ante. We cannot characterize the precise conditionsunder which the market always chooses efficient collateral requirements—or moregenerally, any particular complete or incomplete set of securities—or when there is awelfare-improving role for government, but we do show that government action can

6 This seems relevant to a proper understanding of the history of US housing and mortgage markets. BeforeWorld War I, mortgage down-payment requirements were typically on the order of 50 %. The rise of Savingsand Loan institutions, later the VHA and FHA—and most recently the subprime mortgage market—haveall made it easier for (some) consumers to obtain mortgages with much lower down-payment requirements.Lower down-payment requirements increase competition and drive up housing prices, so some (perhapsvery substantial) portion of the boom in housing prices may have over this period should presumably beascribed to these institutional changes in mortgage markets, rather than to a change in fundamentals. Forcontrast, see Mankiw and Weil (1989).7 For a detailed discussion of the connection between leverage and default, see Fostel and Geanakoplos(2013a).

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be welfare-improving only by taking actions that alter terminal prices (Theorem 4).As long as future prices do not change, no change in lending requirements or produc-tion could benefit everyone. Hence, any valid welfare-based argument for regulationof down-payment requirements would seem to require that regulators could correctlyforecast the price changes that would accompany such regulation.

Following a brief discussion of related literature (below), Sect. 2 presents the modeland Sect. 3 presents the existence theorem (Theorem 1). Section 4 identifies via The-orem 2 the distortion when CE differs from GEI as arising from a liquidity valueand collateral value and shows that efficient CE is Walrasian (Theorem 3). Our simplemortgage market (Example 1) is presented in Sect. 5, and the variants with uncertaintyand default (Examples 2, 3) are presented in Sect. 6. Section 7 shows that, at least insome circumstances, the market chooses the asset structure—in particular the collat-eral requirements—efficiently (Theorem 4). Section 6 concludes. The (long) proof ofTheorem 1 (existence) is relegated to the Appendix.

1.1 Literature

Hellwig (1981) provides the first theoretical treatment of collateral and default in amarket setting; the focus of that work is on the extent to which the Modigliani–Millerirrelevance theorem survives the possibility of default. Dubey et al. (1995); Geanakop-los (1997) and Geanakoplos and Zame (1997, 2002) (the last of which are forerunnersof the present work) provide the first general treatments of a market in which deliv-eries on financial securities are guaranteed by collateral requirements. Geanakoplos(1997) showed how the possibility of default and the need to hold collateral leads toan endogenous choice of securities. The seller of a security is obliged to hold collat-eral that he might like less than the price he has to pay for it, and this inconveniencehinders many security markets (especially Arrow securities) from becoming active;by explaining which securities will not be traded because of the scarcity of collateral,one explains which are. Araujo et al. (2002) use a version of our collateral models toshow that collateral requirements rule out the possibility of Ponzi schemes in infinite-horizon models, and hence eliminate the need for the transversality requirements thatare frequently imposed (Magill and Quinzii 1994; Hernandez and Santos 1996; Levineand Zame 1996). Araujo et al. (2005) expand the model to allow borrowers to set theirown collateral levels, and Steinert and Torres-Martinez (2007) expand the model toaccommodate security pools and tranching.

Dubey et al. (2005) is a seminal work in a somewhat different literature, whichtreats extra-economic penalties for default. (In that particular paper, extra-economicpenalties are modeled as direct utility penalties; when penalties are sufficiently severethat model reduces to the standard model in which enforcement is perfect—and cost-less, because penalties are never imposed in equilibrium.) Default again leads themarket to endogenously choose which securities to trade; a seller who defaults mightbe discouraged from selling because in addition to delivering goods he must deliverpenalties. Another central point of that paper, and of Zame (1993), which uses a verysimilar model, is that the possibility of default may promote efficiency (a point that ismade here, in a different way, in Example 2). Kehoe and Levine (1993) builds a modelin which the consequences of default are exclusion from trade in subsequent financial

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markets, but these penalties constrain borrowing in such a way that there is no equilib-rium default. Sabarwal (2003) builds a model which combines many of these features:securities are collateralized, but the consequences of default may involve seizure ofother goods, exclusion from subsequent financial markets and extra-economic penal-ties, as well as forfeiture of collateral. Kau et al. (1994) provide a dynamic modelof mortgages as options, but ignore the general equilibrium interrelationship betweenmortgages and housing prices.

Geanakoplos (2003) argued that as leverage rises and falls, asset prices will riseand fall in a leverage cycle. Fostel and Geanakoplos (2008) introduced the conceptsof collateral value and liquidity wedge and showed that they necessarily appeared ina simpler model of CE. They also discussed “flight to collateral” as an alternative to“flight to quality”. The notion of liquidity value appears here for the first time.

Bernanke et al. (1996) and Holmstrom and Tirole (1997) are seminal works in aquite different literature that focuses on asymmetric information between borrowersand lenders as the source of borrowing limits. Kiyotaki and Moore (1997) is anotherseminal and influential paper in the macro-literature; it presents a dynamic exampleof a collateral economy.

A substantial empirical literature examines the effect of bankruptcy and default rules(especially with respect to mortgage markets) on consumption patterns and securityprices. Lin and White (2001); Fay et al. (2002); Lustig and Nieuwerburgh (2005) andGirardi et al. (2008) are closest to the present work.

2 Model

As in the canonical model of securities trading, we consider a world with two dates;agents know the present (date 0) but face an uncertain future (date 1). At date 0, agentstrade a finite set of commodities and securities. Between dates 0 and 1, the state ofnature is revealed. At date 1, securities pay off and commodities are traded again.

2.1 Time and uncertainty

There are two dates 0 and 1, and S possible states of nature at date 1. We frequentlyrefer to 0, 1, . . . , S as spots.

2.2 Commodities, spot markets, and prices

There are L ≥ 1 commodities available for consumption and trade in spot marketsat each date and state of nature; the commodity space is R

L(1+S) = RL × R

L S . Abundle x ∈ R

L(1+S) is a claim to consumption at each date and state of the world. Forx ∈ R

L(1+S) and indices s, �, xs is the bundle specified by x in spot s and xs� is thequantity of commodity � specified in spot s. We write δs� ∈ R

L for the commoditybundle consisting of one unit of commodity � in spot s and nothing else. If x ∈ R

L ,then (x, 0) ∈ R

L(1+S) is the bundle in which x is available at date 0 and nothing isavailable at date 1. Similarly, if (x1, . . . , xS) ∈ R

L S , then (0, (x1, . . . , xS)) ∈ RL(1+S)

is the bundle in which xs is available in state s (for each s ≥ 1) and nothing is available

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at date 0. We write x ≥ y to mean that xs� ≥ ys� for each s, �; x > y to mean thatx ≥ y and x �= y; and x � y to mean that xs� > ys� for each s, �.

We depart from the usual intertemporal models by allowing for the possibility thatgoods are durable. If x0 ∈ R

L is consumed (used) at date 0, we write F(s, x0) = Fs(x0)

for what remains in state s at date 1. We assume the map F : S×RL → R

L is continu-ous and is linear and positive in consumption. We denote (F1(x0), . . . , FS(x0)) ∈ R

L S

by F(x0). The commodity 0� is perishable if F(s, δ0�) = 0 for each s ≥ 1 anddurable otherwise. It may be helpful to think of F as being analogous to a productionfunction—except that inputs to production are also consumed.

For each s, there is a spot market for consumption at spot s. Prices at each spotlie in R

L++, so RL(1+S)++ is the space of spot price vectors. For p ∈ R

L(1+S), ps is thevector of prices in spot s and ps� is the price of commodity � in spot s.

2.3 Consumers

There are I consumers (or types of consumers). Consumer i is described by a con-sumption set, which we take to be R

L(1+S)+ , an endowment ei ∈ R

L(1+S)+ , and a utility

function ui : RL(1+S)+ → R.

2.4 Collateralized securities

A collateralized security (security for short) is a pair A = (A, c); A : S ×RL(1+S)++ →

R+ is a continuous function, the promise or face value (denominated in units ofaccount) and c ∈ R

L+ is the collateral requirement. In principle, the promise in states may depend on prices ps in state s and prices p0 at date 0 and even on prices ps′ inother states. The collateral requirement c is a bundle of date 0 commodities; an agentwishing to sell one share of (A, c) must hold the commodity bundle c. By selling asecurity, an agent is effectively borrowing the price while promising the security’sface value. Thus, we sometimes use the words security and loan interchangeably. Theterm security emphasizes that we are assuming a perfectly competitive world in whichlenders and borrowers meet in large markets, and not a world with a single lender andborrower negotiating with each other.

In our framework, the collateral requirement is the only means of enforcingpromises. (Such loans are frequently called no recourse loans.) Hence, if agents opti-mize, the delivery rate or delivery per share of security (A, c) in state s will not bethe face value A(s, p) but rather the minimum of the face value and the value of thecollateral in state s:

Del ((A, c), s, p) = min {A(s, p), ps · F(s, c)}

We take as given a family of J securities A = {(A j , c j )}. (The number J of securitiesmight be very large.) The total delivery on a portfolio θ = (θ1, . . . , θJ ) ∈ R

J is

Del(θ, s, p) =∑

j

θ j Del((A j , c j ); s, p

)

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Because deliveries never exceed the value of collateral, we assume without loss ofgenerality that F(s, c j ) �= 0 for some s. (Securities that fail this requirement willdeliver nothing; in equilibrium, the price of such securities will be 0 and trade in suchsecurities will be irrelevant.) Because sales of securities must be collateralized butpurchases need not be, it is notationally convenient to distinguish between securitypurchases and sales; we writeϕ,ψ ∈ R

J+ for portfolios of security purchases and sales,respectively.8 We assume that buying and selling prices for securities are identical;we write q ∈ R

J+ for the vector of security prices. An agent who sells the portfolioψ ∈ R

J+ will have to hold (and will enjoy) the collateral bundle Coll(ψ) = ∑ψ j c j .

Our formulation allows for nominal securities, for real securities, for options, and forcomplicated derivatives. For ease of exposition, our examples focus on real securities.

2.5 The economy

An economy (with collateralized securities) is a tuple E = 〈({ei , ui )}, {(A j , c j )}〉,where {(ei , ui )} is a finite family of consumers and {(A j , c j )} is a family of collater-alized securities. (The set of commodities and the durable goods technology are fixed,so are suppressed in the notation.) Write e = ∑

ei for the social endowment. Thefollowing assumptions are always in force:

• Assumption 1 e + (0, F(e0)) � 0• Assumption 2 For each consumer i : ei > 0• Assumption 3 For each consumer i :

– (a) ui is continuous and quasi-concave– (b) if x ≥ y ≥ 0 then ui (x) ≥ ui (y)– (c) if x ≥ y ≥ 0 and xs� > ys� for some s �= 0 and some �, then ui (x) > ui (y)– (d) if x ≥ y ≥ 0, x0� > y0�, and commodity 0� is perishable, then ui (x) >

ui (y)

The first assumption says that all goods are represented in the aggregate (keepingin mind that some date 1 goods may only come into being when date 0 goods areused). The second assumption says that individual endowments are nonzero. Thethird assumption says that utility functions are continuous, quasi-concave, weaklymonotone, strictly monotone in date 1 consumption of all goods and in date 0 con-sumption of perishable goods.9

2.6 Budget sets

Given a set of securities A, commodity prices p, and security prices q, a consumerwith endowment e must make plans for consumption, for security purchases and sales,

8 In principle, agents might go long and short in the same security, although there is no reason why theyshould do so and equilibrium would not change whether they did so or not.9 We do not require strict monotonicity in durable date 0 goods because we want to allow for the possibilitythat claims to date 1 consumption are traded at date 0; of course, such claims would typically provide noutility at date 0.

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and for deliveries against promises. In view of our earlier comments, we assume thatdeliveries are precisely the minimum of promises and the value of collateral, so wesuppress the choice of deliveries. We therefore define the budget set B(p, q, e,A) tobe the set of plans (x, ϕ, ψ) that satisfy the budget constraints at date 0 and in eachstate at date 1 and the collateral constraint at date 0:

• at date 0

p0 · x0 + q · ϕ ≤ p0 · e0 + q · ψx0 ≥ Coll(ψ)

In words: expenditures for date 0 consumption and security purchases do not exceedincome from endowment and from security sales, and date 0 consumption includescollateral for all security sales.

• in state s

ps · xs + Del(ψ, s, p) ≤ ps · es + ps · Fs(x0)+ Del(ϕ, s, p)

In words: expenditures for state s consumption and for deliveries on promises donot exceed income from endowment, from the return on date 0 durable goods, andfrom collections on others’ promises.

If these conditions are satisfied, we frequently say that the portfolio (ϕ, ψ) finances xat prices p, q.10

Note that if security promises in each state depend only on commodity prices inthat state and are homogeneous of degree 1 in those commodity prices—in particular,if securities are real (promise delivery of the value of some commodity bundle)—then budget constraints depend only on relative prices. In general, however, budgetconstraints may depend on price levels as well as on relative prices.

2.7 Collateral equilibrium

A collateral equilibrium for the economy E = 〈(ei , ui ),A〉 consists of commodityprices p ∈ R

L(1+S)++ , security prices q ∈ R

J+ and consumer plans (xi , ϕi , ψ i ) satisfyingthe usual conditions:

• Commodity Markets Clear11

∑xi =

∑ei +

∑ (0, F(ei

0))

10 Agents know date 0 prices but must forecast date 1 prices. Our equilibrium notion implicitly incorporatesthe requirement that forecasts be correct, so we take the familiar shortcut of suppressing forecasts andtreating all prices as known to agents at date 0. See Barrett (2000) and Simsek (2013) for models in whichforecasts/beliefs might be incorrect/different.11 As in a production economy, the market-clearing condition for commodities incorporates the fact thatsome date 1 commodities come into being from date 0 activities.

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• Security Markets Clear

∑ϕi =

∑ψ i

• Plans are Budget Feasible

(xi , ϕi , ψ i

)∈ B

(p, q; ei ,A

)

• Consumers Optimize

(x, ϕ, ψ) ∈ B(

p, q, ei ,A)

⇒ ui (x) ≤ ui (xi )

2.8 WE with durable goods

As noted in the Introduction, it is useful to compare/contrast CE with WE and GEI.Here and in the next subsection, we record the formalizations of the latter notions inthe present durable goods framework. We maintain the fixed structure of commoditiesand preferences; in particular, date 0 commodities are durable and F(s, x0) is whatremains in state s if the bundle x0 is consumed at date 0.

A durable goods economy is a family 〈(ei , ui )〉 of consumers, specified by endow-ments and utility functions. We use notation in which a purchase at date 0 conveysthe rights to what remains at date 1; hence if commodity prices are p ∈ R

(1+S)L++ , the

Walrasian budget set for a consumer whose endowment is e is

BW (e, p) ={

x ∈ RL(1+S)+ : p · x ≤ p · e + p · (0, F(x0))

}

A Walrasian equilibrium consists of commodity prices p and consumption choices xi

such that

• Commodity Markets Clear

∑xi =

∑ei +

∑ (0, F(ei

0))


xi ∈ BW(

ei , p)


yi ∈ BW(

ei , p)

⇒ ui(

yi)

≤ ui(

xi)

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2.9 GEI with durable goods

In the familiar GEI model, as in our collateral model, goods are traded on spot markets,but only securities are traded on intertemporal markets. In the GEI context, a securityis a claim to units of account at each future state s as a function of prices; D :S × R

L(1+S) → R. A GEI economy is a tuple 〈(ei , ui ), {D j }〉 of consumers andsecurities.

To maintain the parallel with our collateral framework, we keep security purchasesand sales separate. Given commodity spot prices p ∈ R

L(1+S)++ and security prices

q ∈ RJ , the budget set BGEI(p, q, e, {D j }) for a consumer with endowment e consists

of plans (x, ϕ, ψ) (x ∈ RL(1+S)+ is a consumption bundle; ϕ,ψ ∈ R

J+ are portfoliosof security purchases and sales, respectively) that satisfy the budget constraints at date0 and in each state at date 1:

• at date 0

p0 · x0 + q · ϕ ≤ p0 · e0 + q · ψ

• in state s

ps · xs +∑

j

ψ j D js (p) ≤ ps · es + ps · Fs(x0)+

∑

j

ϕ j D js (p)

Note that the GEI budget set differs from the collateral budget set in that there is nocollateral requirement at date 0 and security deliveries coincide with promises.

A GEI equilibrium consists of commodity spot prices p ∈ RL(1+S)++ , security prices

q ∈ RJ , and plans (xi , ϕi , ψ i ) such that:

• Commodity Markets Clear

∑xi =

∑ei +

∑ (0, F(ei

0))

• Security Markets Clear

∑ϕi =

∑ψ i


(xi , ϕi , ψ i

)∈ B

(ei , p, q,

{D j

})


(x, φ, ψ) ∈ B(

ei , p, q,{

D j})

⇒ ui (x) ≤ ui(

xi)

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3 Existence of collateral equilibrium

Under the maintained assumptions discussed in Sect. 2, CE always exists; we relegatethe proof to the Appendix.

Theorem 1 (Existence) Under the maintained assumptions, every economy admits aCE.

Because we allow for real securities, options, derivatives, and even more compli-cated nonlinear securities, the proof must deal with a number of issues of varyingdegrees of subtlety. Because these issues also arise in standard models, where theycan lead to the non-existence of equilibrium (see Hart (1975) for the seminal exampleof non-existence of equilibrium with real securities, Duffie and Shafer (1985, 1986)for generic existence with real securities, and Ku and Polemarchakis (1990) for robustexamples of non-existence of equilibrium with options), it is useful to understand thesimilarities and especially the differences in our CE framework.

The discussion is most easily presented in the context of a concrete example. Look-ing ahead to the framework of Example 1, consider a world with no uncertainty (S = 1).There are two goods at each date: food F which is perishable and housing H whichis perfectly durable (so that one unit of food at date 0 yields nothing at date 1, while 1unit of housing at date 0 yields one unit of housing at date 1). There is a single security(A, c) which promises A = (p1H − p1F )

+, the difference between the date 1 priceof housing and the date 1 price of food, if that difference is positive and 0 otherwise,and is collateralized by one unit of date 0 housing c = δ0H . (We make assumptionsabout consumer endowments below; consumer preferences will not enter the presentdiscussion.)

The first issue concerns the possibility of unbounded arbitrage. Suppose the com-modity prices are such that p1H − p1F > 0 (so that the promise is strictly positive) butthat q = q(A,c) > p0H . In that case, every consumer could short the security an arbi-trary amount, use the proceeds to buy the required collateral, and have money left overto buy additional consumption—so there would be an unbounded arbitrage, whichwould be inconsistent with equilibrium. Of course this particular unbounded arbitragewould not exist if q < p0H ; the point is only that arbitrage must be ruled out andthat whether or not there is an arbitrage in our model depends on both security pricesand commodity prices, so that the issue is a bit more subtle than in the standard GEImodels. Our proof solves the problem by considering an auxiliary economy in whichwe impose artificial bounds on portfolio choices (these bounds rule out unboundedarbitrage) and then showing that, at equilibrium, these bounds do not in fact bind.12

The second issue concerns a security whose promise is 0. The presence of sucha security whose promise is identically 0 would cause no problems: setting its priceand volume of trade to 0 could not materially affect equilibrium. However, whetheror not the promise A above is 0 depends endogenously on commodity prices, so the

12 In a model with a continuum of agents, any prespecified bounds might bind at equilibrium; we wouldthen have to consider the limit of auxiliary economies as the bounds are relaxed to go to infinity, but theargument would go through using arguments that are familiar in the analysis of economies with a continuumof consumers.

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list of potential prices q must take this into account.13 The problem of 0 prices alsoarises in standard GEI models that admit securities whose promises are allowed tobe negative (in some states), since the equilibrium prices of such securities could bepositive, negative, or zero. We find it convenient to solve the problem in our contextby solving for equilibrium in auxiliary economies in which security promises areartificially bounded away from 0 and then passing to the limit as the artificial limitis relaxed to go to 0 but other approaches, such as those used in the standard GEIliterature, could be used as well.

The third and most serious issue concerns the behavior of budget sets at prices whenp1H = p1F . To illustrate the problem, suppose p0F = p1F = 1, p0H = 2, p1H =1 + ε and that q = q(A,c) = ε, where ε ≥ 0 . For ε > 0, a consumer can use (A, c)to shift wealth from date 0 to date 1 or vice versa. For example, a consumer withendowment (e0F , e0H , e1F , e1H ) = (1, 0, 0, 0) could sell one unit of date 0 food, buy1/ε shares of the security (A, c), collect the proceeds (1/ε)ε = 1 and buy 1 unit of date1 food, obtaining the consumption (x0F , x0H , x1F , x1H ) = (0, 0, 1, 0). However whenε = 0, the promise A = 0 and the consumer can only shift wealth from date 0 to date 1by purchasing date 0 housing; at the given prices, the largest possible consumption ofdate 1 food is x1F = 1/2 (obtained by purchasing 1/2 units of date 0 housing and thenselling the resulting 1/2 unit of date 1 housing). In particular, the consumer’s budgetset is discontinuous at ε = 0. As the reader will recall, such discontinuities in budgetsets lead to non-generic examples of non-existence in economies with real securities(Hart 1975) and to robust examples of non-existence in economies with options (Kuand Polemarchakis 1990).

However, these discontinuities, which present an insuperable obstacle in the morestandard models cited, do not present an insuperable obstacle in our framework. To seewhy, notice that in order for a consumer to actually buy (rather than just demand) 1/εshares of (A, c), the consumer must find counterparties who are willing to sell an equalnumber of shares. In the absence of collateral requirements, such counterparties wouldface no obstacle so long as ε > 0, since selling 1/ε shares of the security requires onlythe transfer of 1 dollar from future wealth to current wealth. In our framework, however,selling 1/ε shares of the security also requires holding 1/ε units of date 0 housing;this will be impossible if ε is small enough that 1/ε exceeds the aggregate supplyof housing. Hence, the discontinuity in the budget should not “bind” at a candidateequilibrium. As above, this idea is most easily carried through in an auxiliary economyin which we impose an artificial bound on security sales and purchases. In this auxiliaryeconomy, there is no discontinuity in demand so an equilibrium exists; if the artificialbound is sufficiently large (in comparison with the aggregate supply of collateral),it does not bind at equilibrium of the auxiliary economy, so an equilibrium for the

13 If equilibrium prices are such that p1H − p1F = 0, then it must also be the case that q = 0. To seethis note that if q > 0, then no one would be willing to buy it but every consumer who held date 0 housingwould wish to sell it, so supply could not equal demand. If q = 0, then trade in (A, c) might occur—andbe indeterminate—but would have no real effects. Note, however, that if the collateral requirement weredifferent, say c′ = δ0F + δ0H , then the equilibrium price of (A, c′) might be positive even if the promiseA = 0 because no consumer would wish to hold both date 0 food and date 0 housing and hence no consumercould sell (A, c′), and of course no one would be willing to buy it.

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auxiliary economy is also an equilibrium for the true economy.14 Note that a similarargument would not work in a standard model in which sales of securities do notneed to be collateralized, because the artificial bounds might bind in every auxiliaryeconomy and the discontinuity would reappear at the candidate equilibrium of thetrue economy. Indeed this is exactly what happens (non-generically) in economieswith real securities and robustly in economies with options.

It may be worth noting that the discontinuity could recur in a way that seemsunavoidable if we expand the model to allow for an infinite set of securities. Suppose,for example, that for each j = 1, . . . there is a security (A j , c j ) whose promise isA j = j (p1H − p1F )

+ and is collateralized by a single house c j = δ0H . Fix j andsuppose prices are p0F = p1F = 1, p0H = 2, p1H = 1+1/j and q = q(A,c) = 1/j .At these prices, any consumer wishing to transfer 1 dollar from current wealth tofuture wealth (a lender) could do so by purchasing 1/j units of the security (A j , c j )

and finding counterparties (borrowers) willing to transfer 1 dollar from future wealth tocurrent wealth by selling 1/j units of the security (A j , c j ). In contrast to the previoussituation, however, taking this position would not pose a problem for the counterpartiessince in order to take this position, the counterparties would need to hold only a singleunit of date 0 housing. As 1/j → 0, the lender and the borrower would transactonly in the security (A j , c j ), but when 1/j = 0 neither security transactions nor thecorresponding wealth transfers could take place. In this situation, the discontinuitycan occur at the candidate equilibrium, so equilibrium might not exist.

4 Distortions

Collateral equilibrium that does not reduce to GEI must involve binding credit con-straints. As we will show, if CE does not reduce to GEI, then some agent would borrowmore (sell more securities) at the going prices (interest rates) if he did not have to putup the collateral to get the loan. He does not do so because the collateral price exceedsits marginal utility to him. When collateral matters, it creates both price and consump-tion distortions of a particular kind. We identify the deviation in commodity prices as a“collateral value,” which leads to commodity prices that are always at least as high asfundamental values and sometimes strictly higher, and the deviation in security pricesas a “liquidity value,” which leads to security prices that are always at least as high asfundamental values and sometimes strictly higher.

Throughout this section, we fix an economy E = 〈{(ei , ui )}, {(A j , c j )}〉 and a CE〈p, q, (xi , ϕi , ψ i )〉 for E . To avoid the issues that surround “corner solutions” and tosimplify the analysis, we maintain throughout this section the following assumptionsfor each consumer i :

(a) consumption is nonzero in each spot: xis > 0

14 Again, the argument could be modified along familiar lines to handle a model with a continuum ofconsumers. An alternative argument could be constructed along somewhat different lines: If counterpartiesdemanded 1/ε units of date 0 housing and ε is small, this would drive up the price of housing (collateral)beyond the ability/willingness of counterparties to pay for it and again it could be shown that at the candidateequilibrium the discontinuity in the budget set would not “bind”. We have chosen our approach only becauseit is technically less complicated.

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(b) consumption of date 0 goods not used as collateral is nonzero: xi0 > Coll(ψ i )

(c) the utility function ui is continuously differentiable at the equilibrium consump-tion xi

(We summarize (a), (b) by saying that equilibrium allocations are financially interior.)Note that we do not impose the requirement that consumption of all goods is positive,only that in each state, there must be positive consumption of at least one good that isnot held as collateral. Hypotheses (a) and (b) would be satisfied, for instance, if everyagent consumed a positive amount of some perishable good like food in each state.

Given these maintained assumptions, we define various marginal utilities. For eachstate s ≥ 1 and commodity k, consumer i’s marginal utility for good sk is

MUisk = ∂ui

(xi

)

∂xsk

By assumption, xs �= 0 so there is some � for which xis� > 0; define consumer i’s

marginal utility of income at state s ≥ 1 to be

μis = 1

ps�MUi

s�

(This definition is independent of which �we choose.) Durability means that i’s utilityfor 0k has two parts: utility from consuming 0k at date 0 consumption and utility fromthe income derived by selling what 0k becomes at date 1; hence, we can expressmarginal utility for 0k as:

MUi0k = ∂ui

(xi

)

∂x0k+

S∑

s=1

μis

[ps · Fs

(δ0k

)]

For any bundle of goods y ∈ RL+, and any s ≥ 0, we define

MUisy =

S∑

k=1

MUsk yk

By assumption, there is some � for which xi0� > Coll(ψ i )0�; define consumer i’s

marginal utility of income at date 0 to be

μi0 = 1

p0�MUi

0�

(Again, this definition is independent of which �we choose.) Finally, define consumeri’s marginal utility for the security (A, c) in terms of marginal utility generated byactual deliveries at date 1

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MUi(A,c) =

S∑

s=1

μis Del

((A, c), s, p

)

For each security (A, c) and commodity 0k or commodity bundle y ∈ RL+, we fol-

low Fostel and Geanakoplos (2008) and define the fundamental values, the collateralvalue, and the liquidity value to consumer i as

FVi(A,c) = MUi

(A,c)

μi0

FVi0k = MUi

0k

μi0

FVi0y = MUi

0y

μi0

CVi0k = p0k − FVi

0k

CVi0y = p0 · y − FVi

0y

LVi(A,c) = q(A,c) − FVi

(A,c)

To understand the terminology, note that if we were in the GEI economy in whichthe security deliveries always coincided with promises and selling the security did notrequire holding collateral, then the equilibrium price of any security would alwayscoincide with its fundamental value to each consumer, while the equilibrium price ofeach good would always be at least as high as its fundamental value to each consumerand would be equal to its fundamental value to each consumer who holds it. Thus,in this GEI economy, the fundamental GEI pricing equations would obtain: for eachconsumer i , commodity sk and security j

MUisk

μis

≤ psk (1)

MUisk

μis

= psk if xisk > 0 (2)

MUi(A j ,c j )

μi0

= q j (3)

Hence, the liquidity value of a security and the collateral value of a commodity aremeasures of the price distortion caused (to a particular agent) by the necessity to holdcollateral.15

Any security sale must be accompanied by the posting of collateral, obtained per-haps through a simultaneous purchase. This simultaneous purchase of a good that

15 Note that if xi0k = 0, then consumer i might find that fundamental pricing holds with MUi

0k/μi0 < p0k

even though he also finds a strictly positive collateral value CVi0k > 0.

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serves as collateral for the security sale that helps to finance the purchase is usuallycalled a leveraged purchase. We define the fundamental value of a leveraged purchaseof bundle of goods c at time 0 via the sale of the security (A, c) as the fundamentalvalue of its residual

FVi0c − FVi

(A,c)

The price of the leveraged purchase is the down payment

p0 · c − q(A,c)

With these definitions in hand, we can clarify the relationship between CE and GEIthrough fundamental values.

Theorem 2 (Fundamental values) Under the assumptions maintained in this section,fundamental values of commodities and securities never exceed prices. If some agenti is selling a security j (ψ i

j > 0), then the liquidity value to him is nonnegative andequal to the collateral value to him of the entire bundle that collateralizes the securityp0 · c j − FVi

c j = q j − FVi(A j ,c j )

. Every other security written against the samecollateral has equal or smaller liquidity value. Moreover, exactly one of the followingmust hold:

• (i) Fundamental value pricing holds for all commodities and securities and theCE is a GEI: Each consumer finds that all date 0 commodities he holds and allsecurities are priced at their fundamental values, so collateral values and liquidityvalues are all zero, and 〈p, q, xi , ϕi , ψ i 〉 is a GEI for the incomplete marketseconomy 〈(ei , ui ), {D j }〉 (where D j is the security whose deliveries are D j (s, p) =Del((A j , c j ), s, p)); or

• (ii) Fundamental value pricing fails and the CE is not a GEI: there is a consumeri , a security (A j , c j ) and a commodity 0k such that ϕi

j = 0 (i is not buying the

security), LVi(A j ,c j )

> 0 (i finds a strictly positive liquidity value for the security),

c j0k > 0 (0k is part of the collateral requirement for the security), and CVi

0k > 0 (ifinds a strictly positive collateral value for 0k).

Before beginning the proof of Theorem 2, it is convenient to isolate part of theargument as a lemma

Lemma For each security (A j , c j ) that is traded:

1. The price of (A j , c j ) is equal to the fundamental value to every agent i who buysit.

2. The net price of the leveraged purchase of the bundle of goods c j via the sale ofthe security (A j , c j ) is equal to the fundamental value of its residual to any agenti who buys it:

p0 · c j − q j = FVic j − FVi

(A j ,c j)(4)

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3. The net marginal utility (of the collateral after making the payments on the loan)per dollar of down payment on (A j , c j ) equals the marginal utility of a dollar spentanywhere else by agent i .

μi0 =

MUic j − MUi

(A j ,c j)

p0 · c j − q j

Proof Consider a security (A j , c j ) that is traded at equilibrium and some agent iwho buys it. Agent i can always reduce or increase the amount ϕi

j that he buys by aninfinitesimal fraction ε, moving the resulting revenue into or out of consumption thatis not used as collateral. Because the agent is optimizing at equilibrium, this marginalmove must yield zero marginal utility, which yields (i).

Now consider a security (A j , c j ) that is traded at equilibrium and some agent i whosells it. Agent i can always reduce or increase all his holding of the collateral bundlec j and the amount ψ i

j of the security that he sells by a common infinitesimal fractionε without violating the collateral constraints, moving the resulting revenue into orout of consumption that is not used as collateral. Because the agent is optimizing atequilibrium, this marginal move must yield zero marginal utility. Keeping in mind thatμi

0 is agent i’s marginal utility for income at date 0, it follows that

MUi c j − MUi(A j ,c j)

= μi0

(p · c j − q j

)

Dividing by μi0 yields (ii); dividing by p0 · c j − q j instead yields (iii). �

Proof of Theorem 2 As we have noted, the budget and market-clearing conditions forCE imply those for GEI. Because utility functions are quasi-concave, in order thatthe given CE reduce to GEI, it is thus necessary and sufficient that the fundamentalpricing Eqs. (1), (2), and (3) hold for each consumer i , commodity sk, and security j .If the given CE does not reduce to GEI, then at least one of these equations must fail;we must show that the failure(s) are of the type(s) specified.

Note that the left-hand sides of the fundamental pricing Eqs. (1), (2), and (3) arejust what we have defined as the fundamental values. Because any agent can alwaysconsume less of some good that she does not use as collateral and use the additionalincome to buy more of any good or of any security, both commodity prices and securityprices must weakly exceed fundamental value for every agent.

Now consider a security (A j , c j ) that is sold at equilibrium and some agent i whosells it. Rearranging Eq. (4) in the Lemma above yields

p0 · c j − FVic j = q j − FVi

(A j ,c j)

As we have already noted, commodity prices are always weakly above fundamentalvalues, so

∑Lk=1 p0kc j

k = p0 · c j > FVic j = ∑L

k=1 FVi0kc j

k exactly when p0k > FVi0k

for some commodity 0k for which c j0k > 0. We conclude that agent i finds a liquidity

value for the security (A j , c j ) he sells if and only if he finds a collateral value for

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some commodity that is part of the collateral c j . The price for each good an agentconsumes but does not use entirely as collateral in date 0, or consumes in any spotat date 1, must equal its fundamental value to him. Hence, if no agent i is selling asecurity with a liquidity value, then every good is priced at its fundamental value toevery agent who holds it.

If there do not exist a security (A j , c j ) and agent i who sells (A j , c j ) and finds botha liquidity value and a collateral value, the only remaining distortion possibility is thatthere is some security (A j ′ , c j ′) that is not sold at equilibrium and some agent i whofinds a liquidity value for (A j ′, c j ′). In that case, agent i could have increased his salesof the security while buying the necessary collateral. Hence, there must be a collateralvalue to him of some good in c j ′ (which he might not be holding in equilibrium). Thiscompletes the proof. �

The Fundamental Values Theorem shows that at the interest rates that prevail in aCE, agents might want to borrow more money than they actually do—if only they didnot have to post collateral. This is indicated precisely by a positive liquidity value forsome security, since borrowing is achieved by selling securities (i.e., loans) and thesecurity price defines an interest rate. Agents are constrained from borrowing at theprevailing, attractive interest rates by the inconvenient need to post collateral, and bya positive collateral value which indicates that the collateral price is higher than themarginal utility of the collateral. The theorem has a slightly paradoxical ring to it. Onemight think that agents who are constrained in their borrowing would be forced todemand fewer durable goods and that therefore the prices of durable goods might beless than their fundamental values. But the theorem asserts the opposite, namely thatthe durable goods used as collateral will always sell for more (or at least as much as)their fundamental values. Kiyotaki and Moore (1997) argue that prices of collateralgoods may be below fundamental values—but only if all date 0 goods are pledgedas collateral, a possibility that is ruled out in Theorem 2 by assumption (b), whichenvisages positive consumption of some non-collateral good like food. For relatedwork in similar models, see Grill et al. (2011) and Fostel and Geanakoplos (2012);for related work in other models, see Garleanu and Pedersen (2011); Hindy (1994);Lagos (2010).

The Lemma shows that the rental price of a durable is always equal to the fun-damental value of using it for one period. Suppose the delivery values D j

s (p) ofthe promise A j are equal to the values of the collateral ps · Fs(c j ) in all states.In this case, the leveraged buyer is simply renting the collateral for time 0. Thefundamental value to the leveraged purchase comes exclusively from the consump-tion utility of the collateral at time 0. Applying part (iii) of the Lemma shows thatthe marginal utility per dollar of rental is indeed equal to the marginal utility ofmoney.

4.1 Collateral value and the efficient markets hypothesis

Theorem 2 tells us that there are two possibilities for a CE. The first is that no agentwould choose to sell more of any security even if he/she did not have to put up

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the collateral (but were still committed to the same delivery rates). In this situation,CE reduces to GEI (with appropriately defined securities payoffs) and fundamentalvalue pricing holds. In this situation, the only (but very important) role played bythe collateral requirement is that of endogenizing security payoffs. The second is thatsome agent would choose to sell more of some security if he/she did not have to putup the collateral (but were still committed to the same delivery rates). In that situation,CE does not reduce to GEI and fundamental value pricing fails for at least one agentand one security; moreover, if the same agent is selling that security, then fundamentalvalue pricing fails for at least one durable good as well. (This point is illustrated sharplyin Example 1, Sect. 5.)

The failure of fundamental value pricing highlights that one must be very careful inapplying the general principle that assets with identical payoffs must trade at identicalprices. To the contrary, durable assets—either physical assets or financial assets—that yield identical payoffs can trade at different prices if one asset is more easilyused as collateral. This would seem to be an especially important point in a settingin which some investors are uninformed/unsophisticated. A central implication ofthe Efficient Markets Hypothesis is that, in equilibrium, prices “level the playingfield” for uninformed/unsophisticated investors and so it is not necessary that suchinvestors know or understand everything about an asset because everything relevantwill be revealed by its price. However, as Theorem 2 shows, this is not quite true:an uninformed/unsophisticated investor who buys a house, expecting that the pricereflects only the consumption value and the future return and forgetting that the pricealso reflects its collateral value, may be sadly disappointed if he does not leverage hispurchase by taking out a big loan against the house or the company. Similarly, a hedgefund that would be eager to buy assets if their purchase could be leveraged may beeager to sell them if they could not be.

4.2 Collateral value and overproduction

We have seen that collateral requirements distort consumption decisions, but theymay distort production decisions as well. To see this, expand the model by allowingeach agent i access to a technology Y i

s ⊂ RL in each spot s that enables the agent

to produce any y ∈ Y is in spot s. (As usual, we interpret negative components of y

as inputs and positive components of output. To be sure that equilibrium exists wecan make the usual assumptions on the production technology.) Since intra-periodproduction is by hypothesis instantaneous, every agent i would choose a produc-tion vector yi

s to maximize profits. However, if some goods are better collateral thanother goods, profit maximization might lead to technologically inferior productionchoices.

For instance, suppose that blue houses could be used as collateral, while whitehouses could not be but that blue houses and white houses are otherwise identical (andin particular are perfect substitutes in consumption); suppose further that blue housesrequire an additional coat of blue paint but otherwise require the same productioninputs as white houses. At equilibrium, blue houses will cost more to produce thanwhite houses, the price of blue houses might exceed the price of white houses, and

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the price difference might exceed the difference in production cost, because the bluehouses have an additional collateral value. In that circumstance, only blue houseswould be produced—even though that is socially inefficient.16

Note that government could ameliorate this inefficiency by changing lending lawsso that white houses could serve as collateral. More generally, government mightimprove welfare by changing lending laws so that more physical goods could be usedas collateral, or by creating new goods—government bonds for instance—that couldbe used as collateral.

4.3 Collateral value and credit rationing

If collateral is the only inducement for delivery, so that security deliveries never exceedcollateral payoffs, then the aggregate value of promises traded cannot exceed theaggregate value of collateral. But the desired level of promises might be much higher,as can be seen, for example, in GEI equilibrium of an economy with the same assetpayoffs. How, in CE, are agents (collectively) restrained from making more promises?The answer is not immediately obvious, for no single agent is directly constrainedfrom borrowing more. Indeed, as long as agents are consuming positive amounts ofnon-collateral goods in equilibrium, any one of them could borrow more by buyingadditional collateral and using it to back another promise. The answer is that eachsecurity sale should really be thought of as a purchase of the residual from the attendantcollateral. If the value of desired security promises exceeded the value of collateral,there would be excess demand for the collateral. Collateral prices would rise, includingcollateral values. The premium necessary to pay to hold the collateral eventually wouldhold desired security sales in check. In short, the scarcity premium or collateral valueof the assets serving as collateral limits borrowing. Again, Example 1 in Sect. 5 makesthis point clearly.

4.4 Liquidity value and endogenous security payoffs

Deliveries on promises are altered by collateral in two ways, one obvious and the otherless obvious but even more important. Without any incentive to deliver beyond thecollateral, security payoffs will be shaped to some extent by the collateral, since theyare the minimum of promises and collateral values. For example, if the collateral hasno value in some state s, then there will be no deliveries in state s. But it would becompletely wrong to presume that total security deliveries are equal or proportionalto total collateral payoffs. For one thing, security payoff types may look very differentfrom collateral payoffs. For another, consider a “financial asset” that provides no utilityat time 0 to its owner. There would be no point in using that asset as collateral for a loanthat promises the whole collateral in every state; the owner could just as easily sell the

16 A similar point is made by Kilenthong (2011) but in a context where white and blue houses are alreadyin existence: blue houses sell for more than white houses because they have a collateral value in additionto a consumption value. Fostel and Geanakoplos (2013b) relates collateral and financial innovation toinvestment.

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asset. Similarly, there would be no point to a loan that promised the proportion λ < 1of the collateral in every state: the owner could sell λ < 1 of the collateral instead.Thus, deliveries on securities backed by financial assets will look very different fromthe payoffs of those financial assets.

Once we have redefined each promise by its delivery rate, the question still remains:which promise will be traded? As Geanakoplos (1997) put it, not every promise typeis rationed the same amount: many potential security types are rationed to zero.

The reason so many kinds of marketed promises are not traded is that many potentialloans must compete for the same collateral, and according to the Fundamental Valuetheorem, all the loans with smaller liquidity value than the corresponding collateralvalue will not be actively traded in equilibrium at all—even though they are availableand priced by the market. Such loan types “waste” collateral. Example 3 in Sect. 6makes precisely this point (among others).

4.5 Liquidity value and inefficient security choices

The market “chooses” the actively traded securities guided by the available collateral,and not by which security could create the greatest gains to trade per dollar expended.There is no reason that the security that maximizes gains to trade per dollar would havethe biggest liquidity value. The security with the largest liquidity value per unit of thecollateral, not the largest liquidity value per dollar of the security, will be traded. Forexample, an Arrow-like security (that promises delivery of the entire collateralizingasset in exactly one state) might provide large gains to trade per dollar of the securityyet have smaller liquidity value than some other security that promises payoffs inmany states. The liquidity value of a security must always be less than its marketprice, and if there are many states in which the Arrow security promises zero, thenthe Arrow security price might be low and it might well have a smaller liquidity valuethan some other security. In that circumstance, the other security might completelychoke off trade in the Arrow security, despite providing smaller gains to trade. Again,Example 3 in Sect. 6 illustrates just this point (among others).

4.6 Efficient collateral equilibria are Walrasian

When markets are incomplete, GEI allocations are generically inefficient—Paretosuboptimal—but in those circumstances in which GEI allocations happen to be Paretooptimal they are in fact Walrasian Elul (1999). Since CE coincide with GEI when thereare no distortions, it should not come as a surprise that when CE allocations happento be Pareto optimal they are also Walrasian.

Theorem 3 Assume the given CE allocation is Pareto optimal and that, in additionto the maintained assumptions, assume that there is at least one consumer h whoconsumes a strictly positive amount of every good: xh

s� > 0 for every s, �. Then, theCE allocation is Walrasian. Indeed, if we define prices π by πs� = MUh

s�/μh0 , then

〈π, xi 〉 is a WE for the Walrasian economy 〈(ei , ui )〉.

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Proof There is no loss in assuming that all contracts (A j , c j ) are traded in equilibrium.(Otherwise, simply delete non-traded contracts.) If agent i is buying the contract(A j , c j ), then q j = FVi

j (otherwise i should have bought more or less of this contract).

Whether or not consumer h had been a buyer of this contract, we must have q j = FVhj ,

for otherwise h could “buy” a little of (A j , c j ) or “sell” a little to one of the buyersi of (A j , c j ) (keeping in mind that there must be buyers, since every contract istraded), making or receiving payment of value q j in date 0 goods that i is consumingat date 0, and delivering (in goods i is consuming in equilibrium) in each state s a tinybit more in value than Del((A j , c j ), s, p). (This is feasible because h is consumingstrictly positive amounts of all goods, and so can make the deliveries by reducing hisconsumption.) This would make both h and i better off, which would contradict Paretoefficiency. As in the proof of Theorem 2, it follows that for all goods k, p0k = FVh

0k ≡π0k . And of course from the fact that h is optimizing in the CE and chose positiveconsumption of each good, it must be that ps is proportional to πs for all s ≥ 1.

To see that 〈π, x〉 must be a WE, choose a j ∈ RL(1+S) so that q j = −p0 · a j

0

and Del((A j , c j ), s, p) = ps · a js for all s ≥ 1. Because q j = FVh

j , it follows that

π · a j = 0, and hence that for each agent i, xi ∈ BW (ei , π). Since (xi ) is a Paretoefficient allocation, 〈π, x〉 must be a WE. If any agent i �= h could improve his utility inhis Walrasian budget set, he could improve it with a very small change while spendingstrictly less (since his utility is quasi-concave and monotonic). Since xh >> 0, anddifferentiable, and since πs� = MUh

s�/μh0, agent h could take the opposite of the trade

and also be strictly better off. �

5 A simple mortgage market

In this section, we offer a simple example that illustrates the working of our model.The example, although representing the simplest possible setting (one perishable con-sumption good, one durable collateral good; no uncertainty) in which the collateralrequirement matters, is surprisingly rich. The example illustrates the distortions quan-tified in Sect. 4, in particular the importance of liquidity value and collateral value.It also illustrates the way in which the interaction between the collateral requirementand the wealth distribution has an enormous effect on prices. Examples 2 and 3, whichfollow in Sect. 6, build on the current Example 1. For the convenience of the reader, wego through some of the (surprisingly complicated) calculations in detail; the readerwho wishes to skip these details might wish to focus on Fig. 1 (which provides agraphical depiction of the price of the collateral good as a function of the collateralrequirement and the wealth distribution, showing the way in which the nature of CEchanges in the various parameter regions) and then go directly to Subsect. 5.1 (whichsummarizes some of the main points).

Example 1 Consider a world with no uncertainty (S = 1). There are two goods ateach date: food F which is perishable and housing H which is perfectly durable.There are two consumers (or two types of consumers, in equal numbers); endowmentsand utilities are as follows:

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Fig. 1 Equilibrium regions and date 0 housing prices

e1 = (18 − w, 1; 9, 0) u1 = x0F + x0H + x1F + x1H

e2 = (w, 0; 9, 0) u2 = log x0F + 4x0H + x1F + 4x1H

Consumer 1 finds food and housing to be perfect substitutes and has constant marginalutility of consumption; Consumer 2 likes housing more than Consumer 1, finds date0 housing and date 1 housing to be perfect substitutes, but has decreasing marginalutility for date 0 food. We takew ∈ (0, 18) as a parameter representing different initialdistributions of wealth. In a moment, we shall add another parameter α representingexogenously imposed borrowing constraints. The example illustrates that in CE, theprice of the durable collateral good housing is very sensitive to the distribution ofwealth and to borrowing constraints, ranging from far below the Walrasian price tofar above the Walrasian price. By contrast, in WE, the price of housing is nearlyimpervious to the distribution of wealth in period 0.

As a benchmark, we begin by recording the unique WE 〈 p, x〉, leaving the simplecalculations to the reader. If we normalize so that p0F = 1, then equilibrium prices,consumptions, and utilities are as follows:

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p0F = 1, p1F = 1, p0H = 8, p1H = 4

x1 = (17, 0; 18 − w, 0) u1 = 35 − w

x2 = (1, 1;w, 1) u2 = 8 + w

Consumer 2 likes housing much more than Consumer 1 and is rich in date 1, so,whatever her date 0 endowment, she buys all the date 0 housing and consumes oneunit of food at date 0—borrowing from her date 1 endowment if necessary, and ofcourse repaying if she does so. Note that the distribution of food at date 1 and individualutilities all depend on w but that that consumption of food and housing at date 0 andthe price of housing do not depend on w. Equilibrium social utility is always 43,which is the level it must be at any Pareto efficient allocation in which both agentsconsume food in date 1. (Both agents have constant marginal utility of 1 for date 1food, so utility is transferable in the range where both consume date 1 food.) Whenw < 9, Consumer 2 borrows (so Consumer 1 lends) to finance date 0 consumption;when w > 9, Consumer 2 lends (so Consumer 1 borrows) in order to finance date 1consumption.

In the GEI world, in which securities always deliver precisely what they promiseand security sales do not need to be collateralized, the Walrasian outcome will againobtain when there are at least as many independent securities as states of nature—here,at least one security whose payoff is never 0.

However, in the world of collateralized securities, Walrasian outcomes need notobtain. When w is small, Consumer 2 is poor at date 0 and so would like to borrow—but the amount she can borrow is constrained by the fact that she will never be requiredto repay more than the future value of the collateral. When w is large, Consumer 2 isrich at date 0 and so would like to lend—but the amount she can lend is constrainedby the fact that the borrower would necessarily need to hold collateral.

Using housing to collateralize, its own purchase is leveraging; regulating this lever-aging can be accomplished by setting collateral requirements. To see the macro-economic effects of regulating leverage, we introduce another exogenous parameterα ∈ [0, 4] that specifies the size of the security promise that can be made using a houseas collateral. We assume that only one security (Aα, c) = (αp1F , δ

0H ) is available fortrade; (Aα, c) promises the value of α units of food in date 1 and is collateralized by1 unit of date 0 housing.17

As we shall see, the nature of CE depends on the parametersw, α; Fig. 1 depicts thevarious equilibrium regions and the price of housing as functions of these parameters.Note that even this simplest of settings is quite rich.

Before beginning the calculations (which are perhaps surprisingly delicate), wemake a useful observation: Increasing α enables Consumer 2 to back more borrowing

17 In our formulation, the security promise and collateral requirement are specified exogenously and thesecurity price is determined endogenously. A more familiar formulation would specify the security priceand the down-payment requirement exogenously and have the interest rate (hence the security promise) bedetermined endogenously. Of course, the two formulations are equivalent: the down-payment requirementd, interest rate r , house price p0H , security price qα , and promise α are related by the obvious equations:d = (p0H − qα)/p0H , r = (α − qα)/qα .

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Table 1 Types of equilibriumψ2 = 0 0 < ψ2 < x2

0H 0 < ψ2 = x20H

x20H = 0 Ia Ib Ic

x20H ∈ (0, 1) IIa IIb IIc

x20H = 1 IIIa IIIb IIIc

with the same collateral and hence to buy more housing with borrowed money.18 SinceConsumer 2 loves housing and is rich in the last period, enabling her to borrow makesher better off, all else equal. But all else need not be equal: when all the Consumer 2types borrow, competition will then raise the price of housing. We trace out the effectsof these opposite forces on her welfare by computing equilibrium for each parameterpair (w, α).

Because we compute equilibria via first-order conditions, especially those of Con-sumer 2, it is convenient to classify equilibria according to the quantity of housingand the amount of borrowing capacity exercised by Consumer 2; by definition the bor-rowing capacity ψ2 cannot exceed housing held, so this leads to 9 potential types ofequilibria, as in Table 1—but because the collateral requirement entails thatψ2 ≤ x2

0H ,there are no equilibria of types Ib, Ic so that only 7 types of equilibria are actuallypossible.

For the given functional forms, we shall see that there are no equilibria of type IIb(although there would be equilibria of type IIb for some other functional forms andparameter values). For all the other types, we solve simultaneously for the equilibriumvariables and the region in the parameter space in which an equilibrium of that typeobtains. We find that these regions are disjoint and partition the parameter space andthat there is a unique equilibrium for each parameter pair (w, α). For many of thevariables, the equilibrium values do not depend on the parameters w, α, and we findthese first; then, we sketch the calculations for the remaining equilibrium variables intypes IIc, IIIc, and IIIa, leaving the details and calculations for other types to the reader.We present quite a lot of detail because the calculations are surprisingly complicatedand illustrate well the notions discussed in Sect. 4.

To solve for equilibrium, we begin by normalizing (as we are free to do) so thatp0F = 1. Because (Aα, c) is a real security, we are also free to normalize so thatp1F = 1. In CE, no agent can begin with less wealth in any state s ≥ 1 than his initialendowment. At date 1, the prices and allocations that prevail are those in the standardexchange economy that results after endowments are adjusted to reflect asset deliveries.It follows that Consumers 1 and 2 each consume food in date 1 (x1

1F > 0, x21F > 0),

that Consumer 2 acquires all the housing at date 1, (x11H =0, x2

1H =1), that the priceof housing in period 1 is 4, (p1H = 4), and that the marginal utilities of moneyto both agents at date 1 are one (μ1

1 = μ21 = 1). Because α ∈ [0, 4], the date 1

value of collateral (weakly) exceeds the promise Aα , so Del(Aα, p) = α; hence,MU1

(Aα,c)= MU2

(Aα,c)= α. The marginal utility to Consumer 2 of owning the house

18 As we shall show, p1H = 4 in every equilibrium, so if α > 4 an agent who sells (Aα, c) will default,delivery will be 4 rather than α and the resulting equilibrium will coincide with the equilibrium that wouldprevail with α = 4.

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in date 0 is obviously 8, since she can live in it at both dates. The marginal utility toConsumer 1 of owning the house at date 0 is 5, since he can live in it at date 0 and sellit for 4 units of food in period 1.

We assert that in every equilibrium, no matter whether or to whom the security issold, the security price qα ≥ α. To see this, suppose qα < α. Because Consumer 1’smarginal utility for food is 1 in both dates, optimality of his equilibrium consumptionmeans that it must not be possible for him to shift from food consumption to holding thesecurity, so necessarily x1

0F = 0. But then x20F = 18 so it is possible for Consumer 2 to

make this shift; since Consumer 2’s marginal utility for date 0 food is 1/x20F = 1/18

and her marginal utility for date 1 food is 1, this is a contradiction. So we concludethat qα ≥ α, as asserted.

Next we assert that p0H ≥ 5. If p0H < 5, then Consumer 1 would strictly prefer tobuy date 0 housing rather than date 0 food so optimality implies that x1

0F = 0. SinceConsumer 1 initially owns the entire housing stock and a strictly positive amount ofdate 0 food, he must be spending some of his date 0 income on purchasing the securityat price qα ≥ α, from which (like food) he gets at most one utile per dollar spent (sinceμ1

1 = 1).0 From this contradiction, we conclude that p0H ≥ 5, as asserted.Finally, we assert that in equilibrium Consumer 1 could never be a net borrower

(sell more of the security than he buys), even in cases where he is very poor in state 0and Consumer 2 is very rich. If he did, then Consumer 2 would have to be a net lender,which by the fundamental pricing lemma implies that

1/x20F

1= MU2

0F

p0F= MU2

(Aα,c)

qα= αμ2

1

qα= α

qα

or x20F = qα/α ≥ 1. But then from Consumer 2’s optimization involving a positive

amount of food,

1/x20F

1= MU2

0F

p0F≥ MU2

0H

p0H= 8

p0H

which implies that p0H ≥ 8x20F . If Consumer 1 is borrowing, then (since he began

with all the housing stock) x10F > 0 and μ1

0 = 1. Moreover, in order to borrow, hemust continue to hold housing. He will only desire that if the fundamental value ofthe residual from the leveraged purchase of housing is at least as high as its price,

MU10H − αμ1

1 ≥ μ10(p0H − qα)

5 − α ≥ p0H − qα ≥ 8x20F − αx2

0F = (8 − α)x20F ≥ 8 − α

which is a contradiction. Thus, Consumer 1 cannot be a net borrower in equilibrium,and so we can take ψ1 = ϕ2 = 0.

Summarizing: for allw ∈ (0, 18), all α ∈ [0, 4], and in every equilibrium, we have

p0F = 1, p1F = 1, p1H = 4, p0H ≥ 5, qα ≥ α,ψ1 = 0,

ϕ2 = 0, μ10 = μ1

1 = μ21 = 1 (5)

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Furthermore, if there is trade in the securities market, then Consumer 1 must be thelender and qα = α.

Lastly, we observe that since housing gives higher utility to consumer 2 than toConsumer 1, and since μ1

0 = μ11 = μ2

1 = 1, the only way they could both holdhousing at date 0 is if x2

0F < 1 so that μ20 > 1. But in that case, Consumer 2 would

borrow as much as he could using his housing as collateral. This rules out equilibriaof type IIb and also rules out equilibria of type IIa except in the trivial case α = 0.

With these preliminary observations out of the way, we shall proceed throughcases in which Consumer 2 gets progressively richer and the price of housing getsprogressively higher. We begin by analyzing equilibrium of the first interesting type,IIc, in which Consumer 1 and Consumer 2 both hold housing and Consumer 2 borrowsall he can on his housing. Since Consumer 1 is lending, we showed already that qα = α.Since Consumer 1 could always buy food, obtaining one utile per dollar expended, andsince he does not have any collateral reason to hold housing, we must have p0H ≤ 5.Combined with the above demonstration that p0H ≥ 5, we deduce p0H = 5.

To solve for the remaining equilibrium variables, we use Consumer 2’s date 0 first-order conditions.19 These first-order conditions derive from the equality between thefundamental value to Consumer 2 of the residual of the leveraged purchase and its price.In other words, buying an additional infinitesimal amount ε of housing costs p0Hε,but of this cost qαε = αε can be borrowed by selling α units of the security, using theadditional housing as collateral, so the net payment is only (p0H −qα)ε = (p0H −α)ε.However, doing this will require repaying the loan in date 1, so the additional utilityobtained will not be MU2

0Hε = 8ε but rather (MU20H − α)ε = (8 − α)ε. On the other

hand, selling an additional ε units of food generates income of p0Fε at a utility cost ofMU2

0Fε. Hence, the correct first-order condition for Consumer 2 is not (6), but rather

1

x20F

= MU20F

p0F= MU2

0H − α

p0H − α= 8 − α

5 − α(6)

19 The correct first-order conditions may not be obvious. Because Consumer 2 holds food and housing atdate 0, it might appear by analogy with standard first-order conditions that

MU20F

p0F= MU2

0Hp0H

MU20F

p0F=

MU2(Aα,c)

qα

Consumer 2 enjoys 4 utils from living in the house at each date, so MU20H = 8. In view of our earlier

calculations, it follows from (6) that MU20F = 8/5 and from (6) that MU2

0F = 1, which is nonsense.The error in this analysis is that these are not the correct first-order conditions for Consumer 2. Theyneglect the collateral value for housing and the liquidity value for securities, respectively. Consumer 2 canborrow against date 1 income by selling the security, but selling the security requires holding collateral. Byassumption, at equilibrium x2

0H = ψ2, so Consumer 2 is exercising all of her borrowing power; hence, shecannot hold less housing without simultaneously divesting herself of some of the security and cannot sellmore of the security without simultaneously acquiring more housing.

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Consumer 2’s date 0 budget constraint is

(5 − α)x20H + x2

0F = w (7)

Solving yields

x20F = 5 − α

8 − α, x2

0H = w − 5−α8−α

5 − α

From this, we can solve for all the equilibrium consumptions and utilities and securitytrades

p0H = 5

qα = α

x1 =(

18 − 5 − α

8 − α, 1 − w − 5−α

8−α5 − α

; 9 + α

[w − 5−α

8−α5 − α

]+ 4

[1 − w − 5−α

8−α5 − α

], 0

)

u1 = 32 − w

x2 =(

5 − α

8 − α,w − 5−α

8−α5 − α

; 9 − α

[w − 5−α

8−α5 − α

]− 4

[1 − w − 5−α

8−α5 − α

], 1

)

u2 = 8 + log(5 − α)− log(8 − α)+(

8 − α

5 − α

)w

ϕ1 = ψ2 = x20H = w − 5−α

8−α5 − α

Finally, the region in which equilibria are of type IIc is defined by the requirementthat x2

0H ∈ (0, 1), so

Region IIc ={(w, α) : 5 − α

8 − α< w <

(5 − α)(9 − α)

8 − α

}

In equilibria of type IIIc, x20H = 1 and ψ2/x2

0H = 1, so Consumer 1 no longerholds housing in date 0, and we cannot guess in advance what the price of housingwill be in period 0, but must solve for it along with the other variables. Reasoning asabove, we see that Consumer 2’s date 0 first-order condition and budget constraint are

8 − α

p0H − α= 1

x20F

p0H − α + x20F = w

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Solving yields

p0H = α +(

8 − α

9 − α

)w

qα = α

x1 =(

18 − w

9 − α, 0; 9 + α, 0

)

u1 = 27 + α − w

9 − α

x2 =(

w

9 − α, 1; 9 − α, 1

)

u2 = log

(w

9 − α

)+ 17 − α

ϕ1 = ψ2 = x20H = 1

The region in which equilibria are of type IIIc is determined by the requirementsthat it be optimal for Consumer 2 to borrow the maximum amount possible, whencex0F ≤ 1, and that Consumer 1 not wish to buy housing, whence p0H ≥ 5. Puttingthese together yields

Region IIIc ={(w, α) :

(5 − α

8 − α

)(9 − α) ≤ w ≤ (9 − α)

}

In region IIIa, the security is not traded but Consumer 2 holds all the housing andsome food at date 0, so Consumer 2 must be rich enough at date 0 to buy all thehousing without borrowing, and must be indifferent to trading date 0 food for date 0housing or for the security. Her first-order conditions and budget constraint at date 0are as follows:

MU2(A,c)

qα= MU2

0F

p0F(8)

MU20F

p0F= MU2

0H

p0H(9)

1 · x20F + p0H · 1 = w (10)

Solving yields

p0H = 8w

9, x2

0F = w

9, qα = αw

9

At these prices, Consumer 1 would like to sell the security, but is deterred from doingso by the requirement to hold (expensive) collateral. Equilibrium consumptions andutilities are as follows:

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x1 = (26 − w, 0; 9, 0)

u1 = 35 − w

x2 =(w

9, 1; 9, 1

)

u2 = log(w

9

)+ 17

ϕ1 = ψ2 = 0

Finally, region IIIa is defined by the requirement that Consumer 2 be rich enough tobuy all the housing at date 0:

Region IIIa = {w : w ≥ 9}

We summarize the description of equilibrium in the remaining regions Ia, IIa, and IIbbelow.

• Ia 0 < w ≤ (5 − α)/(8 − α)

p0H = 5

qα = α

x1 = (18 − w, 1; 13, 0)

u1 = 32 − w

x2 = (w, 0; 5, 1)

u2 = logw + 9

ϕ1 = ψ2 = 0

• IIa α = 0, 5/8 < w < 45/8

p0H = 5

qα = α

x1 =(

18 − 5

8, 1 −

(w

5− 1

8

); 13 − 4

(w

5− 1

8

), 0

)

u1 = 32 − w

x2 =(

5

8,w

5− 1

8; 5 +

(w

5− 1

8

), 1

)

u2 = log

(5

8

)+ 8 + 8w

5

ϕ1 = ψ2 = 0

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• IIIb 9 − α < w < 9

p0H = 8

qα = α

x1 = (17, 0; 18 − w, 0)

u1 = 35 − w

x2 = (1, 1;w, 1)

u2 = 8 + w

ϕ1 = ψ2 = 9 − w

α

5.1 Summary of example 1

It may be useful to summarize some of the points illustrated in Example 1, includingthe distortions identified in Sect. 4.

• In WE, the price and ownership of housing in date 0 is independent of the wealthdistribution in period 0. If Consumer 2 is poor in period 0, she borrows againsther (large) endowment in period 0 to buy all the housing in period 0; if she is richin period 0, she buys all the housing from her period 0 endowment and saves theremainder for period 1; in either case, she holds all the period 0 housing and bidsup its price to her marginal valuation: x2

0H = 1, p0H = 8. By contrast, in CE,the price and ownership of housing depend on the wealth distribution in period 0(and on the collateral requirement). If Consumer 2 is poor in period 0, the collateralrequirement limits her borrowing, hence her purchase of housing and its price:x2

0H < 1, p0H = 5; if she is rich in period 1, the collateral requirement limits hersaving, hence drives up the price of housing (because she has nothing else on whichto spend her wealth): at the limit as w → 18, we see that x2

0H = 1, p0H = 16.• Similarly, the price of housing also rises when α rises, that is, when more borrowing

is allowed. Taken together with the point above, we see that, in this sense, assetprices are much more volatile in CE than in WE.

• CE allocations may be inefficient—and in particular, differ from Walrasianallocations—even when financial markets are “complete.” Because Example 1 rep-resents a transferable utility economy, an allocation is Pareto efficient if and only ifthe sum of individual utilities is 43; these allocations are precisely those for whichConsumer 2 holds all the housing in both dates and exactly one unit of date 0food: x2

0H = x21H = 1 and x2

0F = 1. Hence, CE is Pareto efficient exactly whenw ∈ [9 − α, 9]; that is, in Region IIIb and in portions of the boundaries of RegionsIIIa and IIIc. And, as asserted in Theorem 3, wherever CE is efficient, it is equivalentto WE.

• Forw �∈ [5, 9] CE is inefficient no matter what collateral requirement is set; indeed,CE will be inefficient no matter what collateralized securities are available for trade.This is an example of a more general phenomenon; see Geanakoplos and Zame(2013), Theorem 3.

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• As asserted in Theorem 2, in every region where CE �= GEI, there are distortions andsome consumer experiences a collateral value and liquidity value. In Regions IIcand IIIc this is Consumer 2. To see the distortions more concretely, fixw = 7/2. Forα ∈ (0, 2), parameter values are in region IIc, and for α ∈ [2, 4], parameter valuesare in region IIIc, but in both cases, the collateral requirement distorts Consumer2’s consumption choice, leading her to hold “too much” housing, given the price.To see this, compare marginal utilities per dollar for date 0 food and date 0 housing.In Region IIc, we have

MU20F

p0F= 8 − α

5 − α>

8

5= MU2

0H

p0H

while in Region IIIc, we have

MU20F

p0F= 2(9 − α)

7>

16

7(α + 8−α9−α )

= MU20H

p0H

This distortion can be seen in prices as well: to say that Consumer 2’s marginalutility per dollar for date 0 food exceeds her marginal utility per dollar for date 0housing is to say that the price of date 0 housing is too high. Consumer 2 is willingto pay the higher price of date 0 housing because holding housing enables her toborrow; that is, she derives a collateral value from housing as well as a consumptionvalue. Similarly, Consumer 2 finds the marginal utility per dollar for date 0 food tobe higher than the marginal utility of making the payments on the security; the priceof the security is “too high” as well—she experiences a liquidity value.In the portion of Region IIIa wherew > 9, it is Consumer 1 who experiences collat-eral values and a liquidity value, although Consumer 1 neither holds housing nor sellsthe security. In this region, at the prevailing interest rate (security price), Consumer1 would be delighted to borrow (sell the security) but is discouraged from doingso because he would have to hold collateral, which he does not wish to do. In thisregion, the effect of the collateral distortion is to shut down the borrowing/lendingmarket entirely.

• The effects of collateral requirements on welfare are subtle. Again, fix w = 7/2.Increases in α (equivalently, decreases in the down-payment requirement) make itpossible for consumers of type 2 to access more date 1 wealth. For α ∈ [0, 2), thismakes it possible for consumers of type 2 to afford more housing; the net result isPareto improving. For α ∈ [2, 4], however, consumers of type 2 already own allthe available houses, so increasing α only leads to more competition among them,which serves only to drive up the price of date 0 housing (from p0H = 5 whenα = 2 to p0H = 34/5 when α = 4). This price increase makes Consumers of type1 better off but makes Consumers of type 2 worse off.

• In this Example, we insisted that only one security is offered, a mortgage collateral-ized by a single house and promising the value of α units of food. However, nothingwould change if we allowed for various mortgages, each collateralized by a singlehouse (this is just a normalization) but with different promises. The reason is that

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only the security with the largest liquidity value will be traded; this is the (unique)mortgage with the largest promise.

6 Default, crashes, and welfare

Default and crashes are suggestive of inefficiency. As the Examples presented inthis section show, this need not be true: both default and crashes may be welfareenhancing.20 Indeed, as Example 2 shows, levels of collateral that are socially optimalmay lead to default with positive probability.

More generally, there is a link between collateral requirements and future prices.Lower collateral requirements lead buyers to take on more debt; the difficulties of ser-vicing this debt can lead to reduced demand and lower prices—and even to crashes—inthe future. Importantly, such crashes occur precisely because lower collateral require-ments encourage borrowers to take on more debt than they can service. And yet, despitethese crashes, lower collateral requirements may be welfare enhancing.

A final point made by these Examples is that although the set of securities availablefor trade is given exogenously as part of the data of the model, the set of securitiesthat are actually traded is determined endogenously at equilibrium. Thus, we mayview the financial structure of the economy as chosen by the competitive market.As Example 3 shows, if many collateral levels are available, the market may chooselevels of collateral that lead to default with positive probability, and this choice maybe efficient; moreover, even if all possible securities are available for trade, the marketmay choose an incomplete set to actually be traded at equilibrium.

Example 2 (Default and crashes) We construct a variant on Example 1. Rather thanpresent a full-blown analysis as in Example 1, we fix endowments and take only thesecurity promise as a parameter, making it easier to focus on the points of interest.

There are two states of nature and two goods: Food, which is perishable, andHousing, which is durable. There are two (types of) consumers, with endowments andutility functions:

e1 = (29/2, 1; 9, 0; 9, 0)

u1 = x0F + x0H + (1/2) (x1F + x1H )+ (1/2) (x2F + 3x2H )

e2 = (7/2, 0; 9, 0; 5/2, 0)

u2 = log x0F + 4x0H + (1/2) (x1F + 4x1H )+ (1/2)(x2F + 4x2H )

Note the only differences from Example 1 are that Consumer 1 likes housing better instate 2 and Consumer 2 is poor in state 2—the “bad” state.

A single security—a mortgage—Aα = (αp1F , αp2F ; δ0H ), promising the value ofα units of food and collateralized by 1 unit of housing, is available for trade; we take

20 That default may be welfare enhancing is a point that has been made, in different contexts, by Zame(1993); Sabarwal (2003) and Dubey et al. (2005).

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α ∈ [0, 4] as a parameter.21 (Equivalently, we could consider securities that promiseto deliver the value of one unit of food and are collateralized by 1/α units of housing.)We distinguish four regions; in each, there is a unique equilibrium. In Region I, α issufficiently small that Consumer 2 cannot borrow enough to buy all the housing atdate 0, but buys the remaining housing in date 1. In Region II, α is large enough thatConsumer 2 can buy all the housing at date 0 but small enough that she will be ableto honor her promises in both states at date 1 and retain all the housing at date 1. InRegion III, Consumer 2 will honor her promises but will not be able to retain all thehousing. In Region IV, Consumer 2 will default. Finally, at the boundary of RegionsII and III, prices and equilibrium consumptions are indeterminate. The calculations inRegions I, II are almost identical to those in Example 1; the calculations for RegionsIII, IV follow the same method with the appropriate changes to incorporate default.In all regions, we can take the price of food to be 1 in every state s = 0, 1, 2.

• Region I: α ∈ [0, 2)Consumers 1 and 2 both hold date 0 housing; Consumer 2 borrows as much as shecan in date 0 and honors her promises in both states at date 1. In both states at date1, Consumer 2 buys the remaining housing out of her remaining balance of food.Hence, the price of housing in both states is 4, and the marginal utility of a dollarto each consumer is 1 in both states. It follows that qα = α and that p0H = 5. Thekey equations are the marginal condition for Consumer 2 on the residual from theleveraged purchase of the house and the budget equation that says that all incomenot spent on food must be spent on housing:

1/x20F

p0F= 8 − α

5 − α⇒ x2

0F = 5 − α

8 − α

x20H = 7/2 − 5−α

8−α + αx20H

5

Solving yields

x20H = 1 − 7

2(5 − α)+ 1

8 − α

Hence, equilibrium prices and consumptions are as follows:

p = (1, 5; 1, 4; 1, 4)

x1 =(

18 − 5 − α

8 − α, 1 − x2

0H ; 13 + (α − 4)x20H , 0 ; 13 + (α − 4)x2

0H , 0

)

x2 =(

5 − α

8 − α, x2

0H ; 5 − (α − 4)x20H , 1 ;−3

2− (α − 4)x2

0H , 1

)

(Lest date 1 food consumptions seem strange, remember that (α − 4) < 0.) As αincreases (the collateral requirement becomes less stringent) in this range, there is

21 As before, the case α > 4 reduces to the case α = 4.

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no effect on date 0 housing prices p0H = 5 but Consumer 2 is able to borrow morewhich also gives her an incentive to shift date 0 consumption from food to housing,so her date 0 food consumption decreases and her date 0 housing consumptionincreases: x2

0F = 5/8, x20H = 23/40 when α = 0; x2

0F = 1/2, x20H = 1 when

α = 2.• Region II: α ∈ [2, 5/2)

Consumer 2 holds all the housing at both dates; Consumer 2 borrows as much asshe can in date 0 and honors her promises in both states at date 1. Again the twokey equations are the following:

1/x20F

p0F= 8 − α

p0H − α⇒ x2

0F = p0H − α

8 − α

p0H = 7/2 − p0H − α

8 − α+ α

Equilibrium prices and consumptions are as follows:

p =(

1, α +(

8 − α

9 − α

)(7

2

); 1, 4; 1, 4

)

qα = α

x1 =(

18 − 7

2(9 − α), 0; 9 + α, 0; 9 + α, 0

)

x2 =(

7

2(9 − α), 1; 9 − α, 1; 5

2− α, 1

)

As α increases in this range, date 0 housing prices rise but Consumer 2 is able to bor-row more; she is already holding all the housing at date 0 but can now consume morefood as well: x2

0F = 1/2, x20H = 1, p0H = 5 when α = 2, x2

0F = 7/13, x20H =

1, p0H = 71/13 in the limit as α → 2.5.• Boundary between Regions II, III: α = 5/2

Consumer 2 holds all the housing at both dates; Consumer 2 borrows as much asshe can in date 0; Consumer 2 honors her promises in both states at date 1. In thebad state, Consumer 2 holds all the housing and no food so p2H is indeterminate.This makes the crucial equations a bit more complicated, also becauseμ2

2 = 4/p2H

1/x20F

p0F= 4 + .5(4 − α)+ .5(p2H − α) 4

p2H

p0H − α⇒ x2

0F = p0H − α

8 − (.5 + 2p2H)α

p0H = 7/2 − p0H − α

8 −(.5 + 2

p2H

)α

+ α

123


This gives

p0H = α + 7

2

8 − (.5 + 2p2H)α

9 − (.5 + 2p2H)α

= 5

2+ 7

2

27/4 − 5p2H

31/4 − 5p2H

x20F = 7

2

1

9 −(.5 + 2

p2H

)α

= 7

2

1

31/4 − 5p2H

Notice that as p2H falls (in its indeterminate range), x20F rises and p0H falls. Equi-

librium prices and consumptions are as follows:

p =(

1,5

2+

(7

2

) (27p2H − 20

31p2H − 20

); 1, 4; 1, p2H

)

p2H ∈ [3, 4]qα = 5/2

x1 =(

18 − 14p2H

31p2H − 20, 0; 23

2, 0; 23

2, 0

)

x2 =(

14p2H

31p2H − 20, 1; 13

2, 1; 0, 1

)

When α = 5/2 date 0 food consumption is indeterminate at equilibrium, all otherconsumptions are determinate. For the equilibrium with p2H = 4 (which is the limitof equilibria asα converges to 5/2 from below), x1

0F = 317/13, x20F = 7/13, p0H =

71/13; for the equilibrium with p2H = 3 (which is the limit of equilibria as αconverges to 5/2 from above), x1

0F = 1272/73, x20F = 42/73, p0H = 396/73.

Hence, Consumer 1 is better off “just before” the crash and Consumer 2 is betteroff “just after” the crash. Total utility is higher “just after” the crash,

• Region III: α ∈ ( 52 , 3]

Consumer 2 holds all the housing at date 0; Consumer 2 borrows as much as shecan in date 0; Consumer 2 honors her promises in the good state at date 1. In thebad state, the price of housing falls to p2H = 3; Consumer 2 (who has assets ofendowment plus housing = (5/2) + 3) sells the house, repays her debt and thenbuys all the housing she can afford at the price p2H = 3; Consumer 1 holds theremaining housing. The crucial equations are now a bit more complicated becausep2H = 3 and μ2

2 = 4/3:

1/x20F

p0F= 4 + .5(4 − α)+ .5(3 − α) 4

3

p0H − α�⇒ x2

0F = p0H − α

8 − 76α

p0H = 7/2 − p0H − α

8 − 76α

+ α

123



p =(

1, α +(

7

2

) (48 − 7α

54 − 7α

); 1, 4, 1, 3

)

qα = α

x1 =(

18 − 21

54 − 7α, 0; 9 + α, 0; 23

2,

2α − 5

6

)

x2 =(

21

54 − 7α, 1; 9 − α, 1; 0,

11 − 2α

6

)

As α increases in this range, date 0 housing prices rise again but Consumer 2is again able to borrow more and her consumption of date 0 food rises: p0H =396/73, x2

0F = 42/73 when α = 52 , p0H = 129/22, x2

0F = 7/11 when α = 3.• Region IV α ∈ (3, 4]

Consumer 2 holds all the housing at date 0; Consumer 2 borrows as much as shecan in date 0; Consumer 2 honors her promises in the good state at date 1. In thebad state, the price of housing falls to p2H = 3; Consumer 2, who has assets ofendowment plus housing = (5/2)+ 3, delivers the house (which is worth 3) insteadof her promise (which is α > 3), defaults on her debt, and then buys all the housingshe can afford at the price p2H = 3; Consumer 1 holds the remaining housing. Nowthe price of debt is qα = (.5α + .5(3)) and the crucial equilibrium equations are

1/x20F

p0F= 4 + .5(4 − α)+ .5(3 − 3) 4

3

p0H − qα⇒ x2

0F = p0H − 1.5 − α/2

6 − α/2

p0H = 7/2 − p0H − 1.5 − α/2

6 − α/2+ .5α + 1.5 = 5 − p0H − 1.5 − α/2

6 − α/2+ α/2

These give

(7 − α/2)(p0H − α/2) = 5(6 − α/2)+ 1.5 = 63/2 − 5α/2

p0H = 63/2 − 5α/2

7 − α/2+ α/2 = 63 − 5α

14 − α+ α/2

x20F =

63−5α14−α − 1.5

6 − α/2=

42−3.5α14−α

6 − α/2

= 84 − 7α

(14 − α)(12 − α)= 7

14 − α


p =(

1,α

2+

(63 − 5α

14 − α

); 1, 4; 1, 3

)

qα = α + 3

2

123


x1 =(

29

2−

(7

14 − α

), 0; 9 + α, 0; 23

2,

1

6

)

x2 =(

7

14 − α, 1; 9 − α, 1; 0,

5

6

)

After default, the date 0 housing price and Consumer 2’s date 0 food consumptionrise: p0H = 129/22, x2

0F = 7/11 when α = 3 and p0H = 63/10, x20F = 7/10

when α = 4.

We want to make two very important points about this example:

• As α rises past α = 5/2, the price of housing in the bad state falls precipitouslyfrom 4 to 3: there is a crash. The crash occurs despite the fact that all agents areperfectly rational, have perfect foresight, and hold the same beliefs: the low collateralrequirement (equivalently low down-payment requirement) provides incentive forconsumers of type 2 to take on more debt than they can service. Strikingly, the crashoccurs when α < 3—before consumers of type 2 default on their promises. Perfectforesight entails that the crash is rationally anticipated, so it leads to a sudden dropin the price of housing at date 0, from 71/13 = 5.46 to 396/73 = 5.42.

• As in Example 1, this is a transferable utility economy (Consumers 1 and 2 haveconstant and equal marginal utilities for food in the good state 1), so we mayidentify social welfare with the sum of individual utilities. The effects of collateralrequirement on welfare are complicated; direct computation shows that there are anumber of different regimes:

– 0 ≤ α < 2: welfare of both types of consumers is increasing; social welfare isincreasing

– 2 < α ≤ 5/2: welfare of consumers of type 1 is increasing, welfare of consumersof type 2 is decreasing; social welfare is decreasing

– α = 5/2: welfare of consumers of type 1 jumps down, welfare of consumers oftype 2 jumps up; social welfare jumps up

– 5/2 < α ≤ 3: welfare of consumers of type 1 is increasing, welfare of consumersof type 2 is decreasing; social welfare is decreasing

– 3 < α ≤ 4: welfare of consumers of type 1 is increasing, welfare of consumersof type 2 is decreasing; social welfare is increasing

In particular, social welfare is higher after the crash than immediately before, andsocial welfare is higher after default than immediately before, so collateral levelsthat lead to a perfectly foreseen crash or to perfectly foreseen default can be wel-fare enhancing. It is thus hasty to presume that default and crashes are, all thingsconsidered, destructive to welfare.

In our framework, the set of securities available for trade is given exogenously, butthe set actually traded is determined endogenously at equilibrium. Because the formerset might be very large (conceptually, all conceivable securities), we can view thesecurity market structure itself as determined by the action of the competitive market.As we see below, the result can be default at equilibrium (even when securities thatdo not lead to default are available for trade).

123


Example 3 (Which securities are traded?) We maintain the entire structure of Exam-ple 2, except that some arbitrary set {(A j , c j )} of securities is available for trade. Tobe consistent with our framework, we assume the set of available securities is finite,but, at least conceptually, we might imagine that all possible securities are offered.Because only housing is durable, we assume that only housing is used as collateral;there is no loss in normalizing so that c j = δ0H for each j .

In this setting, only Consumer 2 will sell securities (borrow); as Theorem 1 showed,Consumer 2 will sell only that security which offers her the largest liquidity value.(If more than one security offers Consumer 2 the largest liquidity value, Consumer2 might sell any or all of them.) To see this, suppose that at equilibrium Consumer2 sells (A j , c j ) but that LV2

(Ak ,ck )> LV2

(A j ,c j ). Because both securities require the

same collateral, Consumer 2 could sell ε fewer shares of (A j , c j ) and ε more sharesof (Ak, ck)without violating her collateral constraint. The definition of liquidity valuemeans that this change would strictly improve Consumer 2’s utility, which wouldcontradict the requirement that Consumer 2 optimizes at equilibrium.

Because liquidity values depend on the equilibrium prices and consumptions, it isnot in general possible to order a priori the liquidity values of given securities andhence to know which securities will be traded and which will not be. Instead, weanalyze two particularly interesting scenarios.

Suppose first that mortgages with various promises—but no other securities—areavailable. As above, write Aα = (αp1F , αp2F ; δ0H ). We claim that only the mortgagewith the greatest promise (not exceeding 4—the maximum value of housing in date1; hence, the maximum delivery that will be made on any security) will be traded atequilibrium. To show this, it suffices to show that if α < β ≤ 4 then LVα = LV2

Aα<

LV2Aβ

= LVβ . To this end, we estimate marginal utilities of income μ2s in the various

spots, then fundamental values, security prices, then liquidity values.

• In state 1, prices are p1F = 1, p1H = 4 so μ21 = (1/2)1. In state 2, prices are

p2F = 1, p2H ≥ 3 so μ22 ≤ (1/2)4/3. We claim that μ2

0 > 7/6. To see this, notethat μ2

0 is at least as high as the maximum of marginal utility per dollar for foodand marginal utility per dollar for housing. The former strictly exceeds 7/6 unlessx2

0F ≥ 6/7, and the latter weakly exceeds 8/5 if x20H < 1, in which case p0H = 5.

Hence, to establish that μ20 > 7/6, it remains only to consider the case in which

x20F ≥ 6/7 and x2

0H = 1. In this case, we have

p0H = p0H x20H ≤ 7

2+ 4 − x2

0F ≤ 15

2− 6

7= 93

14

so that the marginal utility per dollar for housing is at least 8/(93/14) = 112/93 >7/6, as asserted.

• Because only Consumer 1 buys securities, security prices coincide with expectedactual deliveries. Write αs, βs for the deliveries Aα,Aβ in state s so prices areqα = (1/2)(α1 + α2) and qβ = (1/2)(β1 + β2).

123


• We have shown above thatμ20 > 7/6 and thatμ2

1 = 1/2 andμ22 ≤ 2/3. Using these

estimates and the definitions, we obtain

μ20

(LVβ − LVα

) =[μ2

0qβ −(μ2

1β1 + μ22β2

)]−

[μ2

0qα −(μ2

1α1 + μ22α2

)]

=[μ2

0 (β1 + β2) /2 −(μ2

1β1 + μ22β2

)]

−[μ2

0 (α1 + α2) /2 −(μ2

1α1 + μ22α2

)]

=[(μ2

0/2)

− μ21

][β1 − α1] +

[(μ2

0/2)

− μ22

][β2 − α2]

> (1/12) (β1 − α1)− (1/12) (β2 − α2)

Because the delivery on any security will be the minimum of its promise and thevalue of collateral, it follows that β1 − α1 = β − α and β2 − α2 ≤ β − α. Henceμ2

0(LVβ − LVα) > 0, whence LVβ − LVα > 0, as asserted.

In particular, if A4—the mortgage with the largest promise—is offered, then only thismortgage will be traded, even though this leads to default in equilibrium.

Now suppose that all possible securities are offered. We assert that in equilibriumonly those securities (with collateral δ0H ) whose deliveries are 4 in state 1 and 3 instate 2 will actually be traded and that equilibrium commodity prices and consumptionscoincide with the equilibrium when only the security A4 above is traded. To see this fixan equilibrium. First suppose the security B is traded and that Del(B, 1, p) < 4. Let B′be any security with the same collateral and state 2 promise (hence delivery) as B, butwhich promises (hence delivers) 4 units of account in state 1. Arguing exactly as above,we see that B′ offers Consumer 2 (the only seller of B) a strictly greater liquidity valuethan does B. Hence, Consumer 2 would strictly prefer to sell B′ rather than B, whichwould be a contradiction. We conclude that if B is traded, then Del(B, 1, p) = 4.Now suppose two securities B,B′ are traded that Del(B, 1, p) = Del(B′, 1, p) = 4but that β2 = Del(B, 2, p) < Del(B′, 2, p) = β ′

2. Arguing as before, we see thatConsumer 2’s date 0 first-order conditions require

8 − 12 (4)− μ2

2(β2)

p0H − 12 (4 + β2)

= 1

x0F= 8 − 1

2 (4)− μ22(β

′2)

p0H − 12 (4 + β ′

2)

However, this would entail β2 = β ′2 which would be a contradiction. We conclude

that all securities traded at equilibrium deliver 4 in state 1 and some common β ≤ 4in state 2.

Finally, consider the magnitude of β. If β > 3, we could argue exactly as inExample 2 to show that default would occur in state 2, which would entail that p2H = 3,and hence that actual delivery would be only 3—a contradiction. If β < 2.5, we couldargue as before to show that the equilibrium price p2H = 4, and hence that any securitywhose delivery is 4 in state 1 and β ′ ∈ (β, 3) in state 2 would offer Consumer 2 agreater liquidity value which would be a contradiction. If 2.5 ≤ β < 3, the argumentis a little more delicate, in part because the price p2H might be indeterminate in theinterval [3, 4], but the conclusion would be the same: any security whose delivery is 4

123


in state 1 and β ′ ∈ (β, 3) in state 2 would offer Consumer 2 a greater liquidity value,which again would be a contradiction.

To see the last assertion, note first that if 2.5 ≤ β < 3, then Consumer 2’s behavioris the same as in Example 2: at date 0, she borrows as much as she can to buy allthe housing; in the good state, she delivers the full promise of 4 from her endowmentand holds all the housing; in the bad state, she delivers the promise of β from herendowment of 5/2 and the value p2H of the housing she owns, and uses the remainderof her wealth to buy back as much housing as she can. If β > 2.5, she cannot buyback all the housing so Consumer 1 buys some of it, whence p2H = 3; if β = 2.5the price of housing is indeterminate: p2H ∈ [3, 4]. As in Example 2, equilibrium isdefined by Consumer 2’s first-order conditions and budget constraints; this yields theequations:

1

x20F

= 4 + (1/2) [(4 − 4)+ (p2H − β)(4/p2H )]

p0H − q

p0F x20F + p0H x2

0H = (7/2)+ qx20H

where q is the security price. Since the security delivers 4 in the good state and β inthe bad state its price is q = (1/2)(4 + β). We have normalized p0F = 1; consumer2 buys all the housing at date 0 so x2

0H = 1. Plugging all these in gives

1

x20F

= 6 − (2β/p2H )

p0H − (1/2)(4 + β)

x20F + p0H = (7/2)+ (1/2)(4 + β)

Solving the second equation for p0H , plugging into the first equation and then solvingyields

x20F = (7/2)

7 − (2β/p2H )

1

x20F

= (2/7) [7 − (2β/p2H )]

Since β ∈ [2.5, 3) and p2H ∈ [3, 4], it follows 1/x20F > 10/7. Because μ2

0 ≥ 1/x20F

it follows that μ20 > 10/7 as well. Now we can argue as before to see that any security

that delivers 4 in state 1 and β ′ ∈ (β, 3) in state 2 would yield Consumer 2 a higherliquidity value; since this is a contradiction, we conclude that β = 3, as asserted.

We conclude that only those securities (with collateral δ0H ) whose deliveries are 4 instate 1 and 3 in state 2 will actually be traded. It follows immediately that equilibriumcommodity prices and consumptions coincide with the equilibrium when only thesecurity A4 (see Example 2) is traded.

Note that even though all possible securities are available, the set of securities thatare traded at equilibrium is endogenously incomplete. Note in particular that Arrowsecurities are not traded, even though they are available, because they make extremelyinefficient use of the available collateral.

123


7 Is the security market efficient?

In the scenario analyzed in Example 3, the market chooses socially efficient promises—even though those promises may lead to default. Characterizing situations when themarket does or does not choose socially efficient sets of securities—or at least Paretoundominated sets of securities—seems an important and difficult question, to whichwe do not know the answer. (Indeed, because multiple equilibria are possible, it isnot entirely clear precisely how to formulate the question.) However, we can give anunambiguous answer about Pareto domination in at least one important case: if date1 prices do not depend on the choices of securities.

Theorem 4 (Constrained efficiency) Every set of CE plans is Pareto undominatedamong all sets of plans that:

(a) are socially feasible(b) given date 0 decisions, respect each consumer’s budget set at every state s at date

1 at the given equilibrium prices(c) call for deliveries on securities that are the minimum of the promise and the value

of collateral

Proof Let 〈p, q, (xi , ϕi , ψ i )〉 be an equilibrium, and suppose that (x i , ϕi , ψ i ) is afamily of plans meeting the given conditions that Pareto dominates the equilibriumset of plans. By assumption, all the alternative plans are feasible, meet the budgetconstraints at each state at date 1, and call for deliveries that are the minimum ofpromises and the value of collateral, optimality of the equilibrium plans at prices p, qmeans, therefore, that all the alternative plans (x i , ϕi , ψ i ) fail the budget constraintsat date 0. Because the alternative set of plans is socially feasible, summing overconsumers yields a contradiction. �

A particular implication of constrained efficiency is that prohibiting trade in cer-tain securities—for example, those that are leveraged above some threshold—cannotlead to a Pareto improvement if it does not lead to a change in date 1 prices. Putdifferently: among security structures that lead to the same date 1 prices, the marketchooses efficiently. In an environment or model in which only one good is availablefor consumption at date 1 and the price of that good is taken to be 1, it is tautologi-cal that all security structures lead to the same date 1 prices, so in that situation CEwill always be constrained efficient and the market will always choose the securitystructure efficiently; compare Kilenthong (2011) and Araujo et al. (2012).22

8 Conclusion

Collateral requirements are almost omnipresent in modern economies, but the effectsof these collateral requirements have received little attention except in circumstances

22 Note that constrained efficiency continues to hold even if we add the possibility of instantaneous produc-tion, as discussed in Subsect. 4.2. Thus, although the market may lead investors to produce technologicallyinferior goods—because of their collateral value—this production will be in the social interest, given theset of possible goods that can be used as collateral, and provided that period 1 prices are not affected.

123


where there is actual default. This paper argues that collateral requirements have impor-tant effects on every aspect of the economy—even when there is no default. Collateralrequirements inhibit lending, limit borrowing, and distort consumption decisions.

When all borrowing must be collateralized, the supply of collateral becomes animportant financial constraint. If collateral is in short supply the necessity of usingcollateral to back promises creates incentives to create collateral and to stretch existingcollateral. The state can (effectively) create collateral by issuing bonds that can beused as collateral and by promulgating law and regulation that make it easier to seizegoods used as collateral.23 The market’s attempts to stretch collateral have drivenmuch of the financial engineering that has rapidly accelerated over the last three-and-a-half decades (beginning with the introduction of mortgage-backed securitiesin the early 1970s) and that has been designed specifically to stretch collateral bymaking it possible for the same collateral to be used several times: allowing agents tocollateralize their promises with other agents’ promises (pyramiding) and allowing thesame collateral to back many different promises (tranching). These two innovationsare at the bottom of the securitization and derivatives boom on Wall Street and havegreatly expanded the scope of financial markets. We address many of these issues in acompanion paper Geanakoplos and Zame (2013) that expands the model presented hereto allow for pyramiding, pooling, and tranching. That work characterizes those efficientallocations that can be supported in equilibrium when financial innovation is possiblebut borrowing must be collateralized; a central finding is that robust inefficiency is aninescapable possibility.

The model offered here abstracts away from transaction costs, informational asym-metries, and many other frictions that play an important role in real markets. It alsorestricts attention to a two-date world and so does not address issues such as defaultat intermediate dates. All these are important questions for later work.

9 Appendix: proof of theorem 1

Proof In constructing an equilibrium for E = ((ei , ui ),A)

, we must confront thepossibility that security promises, hence deliveries, may be 0 at some commodity spotprices.24 (An option to buy gold at $400/ounce will yield 0 if the spot price of gold isbelow $400/ounce.) Because of this, the argument is a bit delicate. We construct, foreach ρ > 0, an auxiliary economy Eρ in which security promises are bounded belowby ρ; in these auxiliary economies, equilibrium security prices will be different from0. We then construct an equilibrium for E by taking limits as ρ → 0.

For each s = 0, 1, . . . , S, choose and fix an arbitrary price level βs > 0. (Becausepromises are functions of prices, choosing price levels is not the same thing as choosingprice normalizations, and we do not assert that equilibrium is independent of theprice levels—only that for every set of price levels there exists an equilibrium.) Write10 = (1, . . . , 1) ∈ RL+ and define subsets of the price space:

23 Similarly, state regulations concerning seizure can have an enormous influence on bankruptcies; see Linand White (2001) and Fay et al. (2002) for instance.24 If collateral requirements are not zero and promises are not 0, then deliveries cannot be 0 either.

123


�s ={(ps�) ∈ R

L++ :∑

ps� = βs

}

� = �0 × . . .×�S

Q ={

q ∈ RJ+ : 0 ≤ q j ≤ 2β010 · c j

}

For each ρ > 0, define an security (Aρ j , c j ) whose promise is Aρ j = A j + ρ. LetAρ = {(Aρ1, c1) . . . , (Aρ J , cJ )}. Define the auxiliary economy Eρ = 〈(ei , ui ),Aρ〉,so Eρ differs from E only in that security promises have been increased by ρ in everystate and for all spot prices. We construct equilibria (for the auxiliary economies andthen for our original economy) with commodity prices in � and security prices in Q.

We first construct truncated budget sets and demand and excess demand correspon-dences in this auxiliary economy. By assumption, collateral requirements for eachsecurity are nonzero. Choose a constant μ so large that μc j

� e0 for each j . (Thus, tosellμ units of the security, Aρ j would require more collateral than is actually availableto the entire economy.) For each (p, q) ∈ � × Q and each consumer i , define thetruncated budget set and the individual truncated demand correspondence

Bi0(p, q) =

{π ∈ Bi (p, q, ei Aρ) : 0 ≤ ϕi

j ≤ μI , 0 ≤ ψ ij ≤ μI for each j

di (p, q) ={π = (x, ϕ, ψ) ∈ Bi

0(p, q) : π is utility optimal in Bi0(p, q)

}

(Note that truncated demand exists at every price (p, q), because we bound securitypurchases and sales. Absent such a bound, demands would certainly be undefined atsome prices. For instance, if q j = 2β010 · c j , agents could sell Aρ j for enough tofinance the purchase of its collateral requirement c j , so there would be an unlimitedarbitrage. Bounding security sales bounds the arbitrage.) Write

D(p, q) =∑

di (p, q)

for the aggregate demand correspondence.For each plan π , we define security excess demand and commodity excess demands

zs(π) in each spot:

za(π) = ϕ − ψ ; zs(π) = xs − es

Write z(π) = (z0(π), . . . , zS(π); za(π)) ∈ RL(1+S) × R

J , and define the aggregateexcess demand correspondence

Z : �× Q → RL(1+S) × R

J ; Z(p, q) = z(D(p, q))

It is easily checked that Z(p, q) is non-empty, compact, and convex for each p, q andthat the correspondence Z is upper hemi-continuous. Because consumptions securitysales are bounded, Z is also bounded below. Because utility functions are monotone, afamiliar argument Debreu (1959) shows that Z satisfies the usual boundary condition:

123


||Z(p, q)|| → ∞ as (p, q) → bdy�× Q

(It does not matter which norm we use.)Now fix ε > 0, and set

�ε = {p ∈ � : ps� ≥ ε for each s, �}

Because Z is an upper hemi-continuous correspondence, it is bounded on�ε×Q; set

Zε =

{z ∈ R

L(1+S) × RJ : ||z|| ≤ sup

(p,q)∈�ε×Q||Z(p, q)||

}

Define the correspondence

Fε : �ε × Q × Zε → �ε × Q × Z

ε

Fε(p, q, z) = argmax{(p∗, q∗) · z : (p∗, q∗) ∈ �ε × Q

} × Z(p, q)

For prices (p, q) ∈ �×Q and a vector of excess demands z ∈ RL(1+S)×R

J , (p, q)·zis the value of excess demands. We caution the reader that, in this setting, Walras’ lawneed not hold for arbitrary prices: the value of excess demand need not be 0. We shallsee, however, that the value of excess demand is 0 at the prices we identify as candidateequilibrium prices.

Our construction guarantees that Fε is an upper-hemicontinuous correspondence,with non-empty, compact convex values. Kakutani’s theorem guarantees that Fε hasa fixed point. We assert that for some ε0 > 0 sufficiently small, the correspondencesFε, 0 < ε < ε0 have a common fixed point. To see this, write Gε ⊂ �ε × Q × Z

ε

for the set of all fixed points of Fε; Gε is a non-empty compact set. We show that forsome ε0 > 0 sufficiently small, the sets Gε are nested and decrease as ε decreases;that is, Gε1 ⊂ Gε2 whenever 0 < ε1 < ε2 < ε0.

To see this, note first that security deliveries are bounded, because deliveries neverexceed the value of collateral. Hence, individual expenditures at budget feasible plans(and in particular at plans in the truncated demand set) are bounded, independentof prices (because income from endowments is bounded, security prices and salesare bounded, and security purchases and deliveries are bounded). Choose an upperbound M > 0 on individual expenditures at budget feasible plans. Because commoditydemands are nonnegative, individual excess demands are bounded below; choose alower bound −R < 0 on individual excess demands.

Because excess demand is the sum of individual demands less the sum of endow-ments, it follows that if z ∈ Z(p, q) then

(p, q) · z ≤ M I and zs� ≥ −RI for each commodity s�

A familiar argument (based on the strict monotonicity of preferences) shows thatif commodity prices tend to the boundary of �, then aggregate commodity excessdemand blows up. If the price of some security tends to 0 but the value of its collateral

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does not, then deliveries on that security do not tend to 0, whence demand for thatsecurity and consequent aggregate commodity excess demand again blow up. Hence,we can find ε0 > 0 such that if (p, q) ∈ � × Q, z ∈ Z(p, q), and ps0�0 < ε0 forsome spot s0 and commodity �0, then there is some spot s1 and commodity �1 suchthat

zs1�1 >1

βs1 − (L − 1)ε

[M I + εR(L − 1)+ (max

sβs)RI

](11)

We assert that if 0 < ε < ε0, then Gε ⊂ �ε0 × Q × Zε0 . To see this, suppose that

(p, q, z) ∈ Gε and p /∈ �ε0 . Define p ∈ � by

ps� =⎧⎨

⎩

ε if s = s0, � �= �0βs − (L − 1)ε if s = s0, � = �0βs/L otherwise

Direct calculation using Eq. (11) shows that ( p, 0) · z > M I , which is a contradiction.We conclude that p ∈ �ε0 and hence that (p, q, z) ∈ Gε

0 as desired.The definition of Fε implies that if 0 < ε1 < ε2 and Gε1 ⊂ �ε2 × Q × Z

ε2 , thenGε1 ⊂ Gε2 . Hence, for 0 < ε < ε0, the sets Gε are nested and decrease as ε decreases.A nested family of non-empty compact sets has a non-empty intersection so we maydefine the non-empty set G:

G =⋂

ε<ε0

Gε

Let (p, q, z) ∈ G; we assert that z = 0 and that p, q constitute equilibrium prices forthe economy Eρ .

We first show that excess security demand z ja = 0 for each j . If z j

a > 0, therequirement that (p, q) maximize the value of excess demand would imply that q j

is as big as possible: q j = 2β010 · c j . But then agents could sell Aρ j for enoughto finance the purchase of the collateral requirement, whence the excess demand forAρ j would be negative, a contradiction. We conclude that security excess demandmust be non-positive. If the excess demand for security j were strictly negative, therequirement that (p, q) maximize the value of excess demand would imply that q j isas small as possible: q j = 0. But if the price of Aρ j were 0, then every agent wouldwish to buy it because its delivery would be min{ρ, ps · Fs(c j )} > 0. Hence, theexcess demand for Aρj must be positive, a contradiction.25 We conclude that za = 0.

We claim that Walras’ law holds: (p, q) · z = 0. To see this, choose individualdemands π i ∈ di (p, q) for which the corresponding aggregate excess demand is z:Z(

∑π i ) = z. For each agent i , the planπ i lies in the budget set at prices (p, q), so the

date 0 expenditure required to carry out the plan π i is no greater than the value of date0 endowment. Because utility is strictly monotone in date 0 perishable commodities

25 Note that we could not obtain this conclusion in the original economy, because at the prices (p, q) thesecurity A j might promise 0 in every state.

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and in all commodities in state s, optimization implies that all individuals spend alltheir income at date 0, so we conclude that the date 0 expenditure required to carryout the plan π i is precisely equal to the value of date 0 endowment; i.e., the value ofdate 0 excess demand is 0 for each individual. Summing over all individuals showsthat the value of date 0 aggregate excess demand is 0: p0 · z0 + q · za = 0. Nowconsider any state s ≥ 1 at date 1. We can argue exactly as above to conclude that thevalue of each individual’s excess demand is equal to the net of deliveries on purchasesand sales of securities. Thus, the value of aggregate excess demand in state s is thenet of deliveries on aggregate purchases and sales of securities. However, za = 0 soaggregate purchases and sales of securities are equal, and so the value of aggregateexcess demand in state s is 0. Summing over all spots, we conclude that (p, q) · z = 0,as asserted.

We show next that z = 0. If not, Walras’ law entails that excess demand for somecommodity is positive; say zs0�0 > 0. Define commodity prices p by:

ps� =⎧⎨

⎩

ps� if s �= s0ε if s = s0, � �= �01 − (L − 1)ε if s = s0, � = �0

Because (p, q) · z = 0 and zs0�0 > 0, ( p, q) · z will be strictly positive if ε is smallenough. However, this would contradict our assumption that (p, q, z) ∈ G and henceis a fixed point of Fε for every sufficiently small ε. We conclude that z = 0. Hence,〈p, q, (π i )〉 is an equilibrium for the economy Eρ .

It remains to construct an equilibrium for the original economy E . To this end,let p(ρ), q(ρ), (π i (ρ)) be equilibrium prices and plans for Eρ and let ρ → 0. Byconstruction, prices and plans lie in bounded sets, so we may choose a sequence(ρn) → 0 for which the corresponding prices and plans converge; let the limits bep, q, (π i ). Commodity prices p do not lie on the boundary of � (for otherwise theexcess demands at prices p(ρn), q(ρn)would be unbounded, rather than 0). It followsthat π i (ρ) is utility optimal in consumer i’s budget set at prices (p, q). Because thecollection of plans (π i ) is the limit of collections of socially feasible plans, it followsthat they are socially feasible and hence that the artificial bounds on security purchasesand sales do not bind at the prices p, q. Hence, 〈p, q, (π i )〉 is an equilibrium breakfor E . �

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collateral equilibrium, i: a basic framework by john

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